753:
130:
31:
5419:
3005:; an ordered set possesses the greatest-lower-bound property if and only if it also possesses the least-upper-bound property; the least-upper-bound of the set of lower bounds of a set is the greatest-lower-bound, and the greatest-lower-bound of the set of upper bounds of a set is the least-upper-bound of the set.
2012:
Finally, a partially ordered set may have many minimal upper bounds without having a least upper bound. Minimal upper bounds are those upper bounds for which there is no strictly smaller element that also is an upper bound. This does not say that each minimal upper bound is smaller than all other
743:
which is smaller than all the positive real numbers and greater than any other real number which could be used as a lower bound. An infimum of a set is always and only defined relative to a superset of the set in question. For example, there is no infimum of the positive real numbers inside the
7806:
5247:
5220:
4877:
8708:
6923:
6798:
5666:
7472:
4997:
9125:
8400:
1918:
For example, consider the set of negative real numbers (excluding zero). This set has no greatest element, since for every element of the set, there is another, larger, element. For instance, for any negative real number
8809:
8964:
8296:
7104:
7225:
3562:
7649:
9938:
9808:
9566:
7327:
4544:
7743:
8890:
8217:
6387:
6562:
6494:
8455:
4765:
4377:
4256:
7550:
4628:
1573:
5465:
4102:
3909:
6263:
6130:
6195:
4302:
4181:
6671:
5892:
5602:
6004:
9744:
8155:
5124:
9849:
7893:
7683:
7393:
4381:
Expressing the infimum and supremum as a limit of a such a sequence allows theorems from various branches of mathematics to be applied. Consider for example the well-known fact from
9228:
7265:
7023:
8002:
5132:
5074:
3654:
4674:
4453:
3424:
3361:
6829:
6704:
5844:
4135:
4060:
3867:
3689:
3512:
3477:
9368:
7943:
7738:
7360:
3622:
2783:
2115:
718:
2992:
2922:
2195:
2166:
1972:
1638:
subsets have both a supremum and an infimum. More information on the various classes of partially ordered sets that arise from such considerations are found in the article on
686:
637:
9971:
9878:
7586:
9434:
4027:
3834:
8987:
8517:
5414:{\displaystyle \|g\|_{\infty }^{q}~{\stackrel {\scriptscriptstyle {\text{def}}}{=}}~\left(\sup _{x\in \Omega }|g(x)|\right)^{q}=\sup _{x\in \Omega }\left(|g(x)|^{q}\right)}
3162:
2597:
3590:
2755:
2392:
2350:
2328:
2217:
2137:
2083:
1617:
1595:
10363:
7505:
5539:
2848:
2670:
9630:
5507:
2004:
Whereas maxima and minima must be members of the subset that is under consideration, the infimum and supremum of a subset need not be members of that subset themselves.
9662:
9250:
8564:
8053:
4909:
1164:
817:
3981:
3788:
8611:
6824:
6699:
5810:
4760:
3191:
3118:
1431:
1262:
1233:
1084:
915:
886:
5962:
5939:
5691:
3932:
3739:
9698:
5775:
5755:
5711:
4576:
2713:
9595:
6952:
9281:
9171:
8584:
8478:
8073:
7154:
7028:
5243:
4476:
3092:
3049:
2892:
2561:
2518:
2475:
1940:
1914:
1887:
1844:
1817:
1794:
1406:
1059:
741:
487:
304:
115:
9412:
9388:
9323:
9303:
9193:
9148:
8604:
8093:
7866:
7846:
7826:
7703:
7131:
6609:
6589:
6429:
6409:
6310:
6290:
6056:
6036:
5916:
5731:
5607:
5094:
5017:
4734:
4714:
4694:
4403:
4322:
4201:
4001:
3952:
3808:
3759:
3716:
3451:
3381:
3322:
3302:
3278:
3250:
3211:
3069:
3026:
2962:
2942:
2869:
2823:
2803:
2733:
2690:
2637:
2617:
2538:
2495:
2452:
2432:
2412:
2370:
2306:
2286:
2266:
2058:
2038:
1992:
1864:
1771:
1743:
1723:
1703:
1683:
1663:
1520:
1500:
1480:
1451:
1383:
1363:
1339:
1305:
1285:
1204:
1184:
1132:
1104:
1036:
1016:
992:
958:
938:
857:
837:
782:
657:
555:
535:
507:
464:
440:
420:
376:
352:
324:
281:
257:
234:
191:
171:
151:
92:
72:
52:
7398:
7159:
4918:
10570:
8302:
10587:
8714:
7270:
8896:
8223:
744:
positive real numbers (as their own superset), nor any infimum of the positive real numbers inside the complex numbers with positive real part.
5964:) can also be used to help prove many of the formula given below, since addition and multiplication of real numbers are continuous operations.
3517:
6315:
8098:
More generally, if a set has a smallest element, then the smallest element is the infimum for the set. In this case, it is also called the
6132:
consisting of all possible arithmetic sums of pairs of numbers, one from each set. The infimum and supremum of the
Minkowski sum satisfies
7591:
1974:
which is greater. On the other hand, every real number greater than or equal to zero is certainly an upper bound on this set. Hence,
9883:
9753:
9503:
4481:
2013:
upper bounds, it merely is not greater. The distinction between "minimal" and "least" is only possible when the given order is not a
10387:
9449:
6200:
6061:
6135:
6614:
1622:
Consequently, partially ordered sets for which certain infima are known to exist become especially interesting. For instance, a
1522:
has no lower bound at all, or if the set of lower bounds does not contain a greatest element. (An example of this is the subset
8815:
8161:
2567:
set also has the least-upper-bound property, and the empty subset has also a least upper bound: the minimum of the whole set.
6499:
2040:
be the set of all finite subsets of natural numbers and consider the partially ordered set obtained by taking all sets from
10081:
6434:
10519:
10170:
2001:
is more general. In particular, a set can have many maximal and minimal elements, whereas infima and suprema are unique.
8407:
10445:
9458:
4327:
4206:
1994:
is the least upper bound of the negative reals, so the supremum is 0. This set has a supremum but no greatest element.
10582:
7801:{\displaystyle x\leq y{\text{ in }}P^{\operatorname {op} }\quad {\text{ if and only if }}\quad x\geq y{\text{ in }}P,}
7510:
5972:
The following formulas depend on a notation that conveniently generalizes arithmetic operations on sets. Throughout,
4581:
1525:
10501:
10089:
10025:
9470:
3193:
For example, it applies for real functions, and, since these can be considered special cases of functions, for real
5424:
4065:
3872:
3281:
4261:
4140:
5852:
5544:
10481:
10461:
5975:
4406:
9714:
8107:
5103:
10577:
10511:
10415:
10298:
10124:
9813:
7871:
7661:
7365:
5215:{\displaystyle \|g\|_{\infty }\,{\stackrel {\scriptscriptstyle {\text{def}}}{=}}\,\sup _{x\in \Omega }|g(x)|}
122:
10114:
9198:
7232:
9464:
1685:
contains a greatest element, then that element is the supremum; otherwise, the supremum does not belong to
7948:
6957:
5026:
4872:{\displaystyle f(\sup S)=f\left(\lim _{n\to \infty }s_{n}\right)=\lim _{n\to \infty }f\left(s_{n}\right),}
3627:
10565:
10435:
10119:
4633:
4412:
3390:
3327:
2238:
1639:
5817:
4114:
4032:
3839:
3659:
3482:
3456:
10529:
10471:
10328:
9332:
7901:
7712:
7332:
3595:
3217:
2760:
2228:
2088:
691:
2967:
2897:
2171:
2142:
1945:
1725:
contains a least element, then that element is the infimum; otherwise, the infimum does not belong to
662:
613:
10534:
10466:
9943:
9854:
7555:
10486:
9417:
4006:
3813:
10560:
10539:
10476:
9128:
8483:
3123:
2577:
3569:
2738:
2375:
2333:
2311:
2200:
2120:
2066:
1600:
1578:
720:
There is, however, exactly one infimum of the positive real numbers relative to the real numbers:
10341:
10163:
7477:
5512:
5020:
2829:
2642:
17:
9600:
8703:{\displaystyle \sup\{x\in \mathbb {R} :0<x<1\}=\sup\{x\in \mathbb {R} :0\leq x\leq 1\}=1.}
5470:
10491:
10420:
10313:
10293:
10069:
9635:
9480:
9257:
9233:
8531:
8020:
4882:
2785:
is the least upper bound, a contradiction is immediately deduced because between any two reals
1137:
790:
603:, but are more useful in analysis because they better characterize special sets which may have
573:
514:
331:
10017:
3957:
3764:
10680:
10675:
10318:
10210:
6803:
6678:
5782:
4739:
3229:
3167:
3097:
1410:
1238:
1212:
1063:
891:
865:
785:
608:
581:
237:
5944:
5921:
5673:
3914:
3721:
10496:
10267:
9747:
9671:
6918:{\displaystyle \inf(r\cdot A)=r(\sup A)\quad {\text{ and }}\quad \sup(r\cdot A)=r(\inf A).}
6793:{\displaystyle \inf(r\cdot A)=r(\inf A)\quad {\text{ and }}\quad \sup(r\cdot A)=r(\sup A),}
5760:
5740:
5696:
4549:
2695:
585:
9571:
6928:
5661:{\displaystyle \sup S\,{\stackrel {\scriptscriptstyle {\text{def}}}{=}}\,\|g\|_{\infty }.}
8:
10652:
10440:
10323:
10262:
10231:
7467:{\displaystyle {\frac {1}{\displaystyle \sup _{s\in S}s}}=\inf _{s\in S}{\tfrac {1}{s}}.}
2014:
9263:
9153:
8569:
8460:
8058:
7136:
5225:
4992:{\displaystyle f(S)\,{\stackrel {\scriptscriptstyle {\text{def}}}{=}}\,\{f(s):s\in S\}.}
4458:
4137:
is any non-empty set of real numbers then there always exists a non-decreasing sequence
3074:
3031:
2874:
2543:
2500:
2457:
1922:
1896:
1869:
1826:
1799:
1776:
1388:
1041:
723:
469:
286:
97:
10622:
10524:
10430:
10377:
10303:
10236:
10156:
9665:
9397:
9373:
9326:
9308:
9288:
9178:
9133:
8589:
8078:
7851:
7831:
7811:
7688:
7116:
6594:
6574:
6414:
6394:
6295:
6275:
6041:
6021:
5901:
5734:
5716:
5079:
5002:
4719:
4699:
4679:
4388:
4307:
4186:
3986:
3937:
3793:
3744:
3701:
3436:
3366:
3307:
3287:
3263:
3235:
3196:
3054:
3011:
2947:
2927:
2854:
2808:
2788:
2718:
2675:
2622:
2602:
2523:
2480:
2437:
2417:
2397:
2355:
2291:
2271:
2251:
2043:
2023:
1977:
1849:
1756:
1728:
1708:
1688:
1668:
1648:
1505:
1485:
1465:
1436:
1368:
1348:
1324:
1290:
1270:
1189:
1169:
1117:
1089:
1021:
1001:
977:
943:
923:
842:
822:
767:
642:
540:
520:
492:
449:
425:
405:
361:
337:
309:
266:
242:
219:
176:
173:(red diamond and circles), and the smallest such upper bound, that is, the supremum of
156:
136:
77:
57:
37:
9120:{\displaystyle \sup\{f(t)+g(t):t\in A\}~\leq ~\sup\{f(t):t\in A\}+\sup\{g(t):t\in A\}}
8606:
is an upper bound, but it is not the least upper bound, and hence is not the supremum.
10410:
10095:
10085:
10021:
10007:
8974:
7706:
2219:
nor is the converse true: both sets are minimal upper bounds but none is a supremum.
10612:
10551:
10372:
10272:
10227:
10215:
10073:
2826:
1631:
260:
10080:. Grundlehren der mathematischen Wissenschaften. Vol. 317. Berlin New York:
8970:
2017:
one. In a totally ordered set, like the real numbers, the concepts are the same.
1998:
1890:
1820:
1623:
1462:
Infima and suprema do not necessarily exist. Existence of an infimum of a subset
8395:{\displaystyle \inf \left\{(-1)^{n}+{\tfrac {1}{n}}:n=1,2,3,\ldots \right\}=-1.}
2241:
which is typical for the set of real numbers. This property is sometimes called
588:. However, the general definitions remain valid in the more abstract setting of
10425:
10241:
8978:
8804:{\displaystyle \sup \left\{(-1)^{n}-{\tfrac {1}{n}}:n=1,2,3,\ldots \right\}=1.}
5097:
4912:
688:
could simply be divided in half resulting in a smaller number that is still in
10134:
10054:
8095:
is a lower bound, but not the greatest lower bound, and hence not the infimum.
10669:
10637:
10632:
10617:
10607:
10308:
10222:
10197:
10099:
8959:{\displaystyle \sup \left\{x\in \mathbb {Q} :x^{2}<2\right\}={\sqrt {2}}.}
6015:
5895:
5778:
1316:
969:
443:
8291:{\displaystyle \inf \left\{x\in \mathbb {Q} :x^{3}>2\right\}={\sqrt{2}}.}
2330:
of all real numbers has the least-upper-bound property. Similarly, the set
10394:
10193:
10002:
7099:{\displaystyle \inf(-A)=-\sup A\quad {\text{ and }}\quad \sup(-A)=-\inf A.}
4258:
Similarly, there will exist a (possibly different) non-increasing sequence
3253:
752:
589:
7220:{\displaystyle {\frac {1}{S}}~:=\;\left\{{\tfrac {1}{s}}:s\in S\right\}.}
3257:
2308:
is said to have the least-upper-bound property. As noted above, the set
1705:(or does not exist). Likewise, if the infimum exists, it is unique. If
577:
118:
3557:{\displaystyle -\infty =\sup \varnothing <\inf \varnothing =\infty .}
129:
10647:
10338:
2998:; there is no least upper bound of the set of positive infinitesimals.
2995:
2564:
30:
10642:
10627:
10139:
9391:
3384:
2168:
are greater than all finite sets of natural numbers. Yet, neither is
3028:
every bounded subset has a supremum, this applies also, for any set
10282:
10251:
10202:
10179:
5847:
5127:
4382:
9253:
8099:
4999:
If in addition to what has been assumed, the continuous function
2061:
600:
596:
7898:
For subsets of the real numbers, another kind of duality holds:
7644:{\displaystyle {\tfrac {1}{\inf _{}S}}=\sup _{}{\tfrac {1}{S}}.}
4455:
is a sequence of points in its domain that converges to a point
3284:
implies (and is equivalent to) that any bounded nonempty subset
9933:{\displaystyle {\tfrac {1}{\inf _{}S}}=\sup _{}{\tfrac {1}{S}}}
9803:{\displaystyle {\tfrac {1}{\sup _{}S}}=\inf _{}{\tfrac {1}{S}}}
9561:{\displaystyle f\left(s_{1}\right),f\left(s_{2}\right),\ldots }
9437:
7322:{\displaystyle {\frac {1}{\sup _{}S}}~=~\inf _{}{\frac {1}{S}}}
4539:{\displaystyle f\left(s_{1}\right),f\left(s_{2}\right),\ldots }
214:
10131:
10034:
6312:
of real numbers is defined similarly to their
Minkowski sum:
5076:
This may be applied, for instance, to conclude that whenever
10148:
10005:(1976). ""Chapter 1 The Real and Complex Number Systems"".
1748:
9329:
of the subsets when considering the partially ordered set
7395:
is used. This equality may alternatively be written as
2540:
and is less than or equal to every other upper bound for
2288:
having an upper bound also has a least upper bound, then
153:
of real numbers (blue circles), a set of upper bounds of
8885:{\displaystyle \sup\{a+b:a\in A,b\in B\}=\sup A+\sup B.}
8212:{\displaystyle \inf\{x\in \mathbb {R} :0<x<1\}=0.}
2894:
which itself would have to be the least upper bound (if
2117:
ordered by subset inclusion as above. Then clearly both
659:) does not have a minimum, because any given element of
561:. Consequently, the supremum is also referred to as the
9707:
9705:
9475:
Pages displaying short descriptions of redirect targets
9454:
Pages displaying short descriptions of redirect targets
6382:{\displaystyle A\cdot B~:=~\{a\cdot b:a\in A,b\in B\}.}
3256:
are particularly important. For instance, the negative
1597:. It has upper bounds, such as 1.5, but no supremum in
592:
where arbitrary partially ordered sets are considered.
9919:
9888:
9859:
9789:
9758:
9719:
9467: – Element that is not ≤ (or ≥) any other element
8749:
8337:
7627:
7596:
7522:
7450:
7186:
6960:
6557:{\displaystyle \sup(A\cdot B)=(\sup A)\cdot (\sup B).}
5627:
5284:
5162:
4944:
3304:
of the real numbers has an infimum and a supremum. If
3260:
do not have a greatest element, and their supremum is
1950:
54:
of real numbers (hollow and filled circles), a subset
10344:
9946:
9886:
9880:
is usually left undefined, which is why the equality
9857:
9816:
9756:
9717:
9674:
9638:
9603:
9574:
9506:
9420:
9400:
9376:
9335:
9311:
9291:
9266:
9236:
9201:
9181:
9156:
9136:
8990:
8899:
8818:
8717:
8614:
8592:
8572:
8534:
8486:
8463:
8410:
8305:
8226:
8164:
8110:
8081:
8061:
8023:
7951:
7904:
7874:
7854:
7834:
7814:
7746:
7715:
7691:
7664:
7594:
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7513:
7480:
7407:
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7335:
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7235:
7162:
7139:
7119:
7031:
6931:
6832:
6806:
6707:
6681:
6617:
6597:
6577:
6502:
6489:{\displaystyle \inf(A\cdot B)=(\inf A)\cdot (\inf B)}
6437:
6417:
6397:
6318:
6298:
6278:
6203:
6138:
6064:
6044:
6024:
5978:
5947:
5924:
5904:
5855:
5820:
5785:
5763:
5743:
5733:
be non-increasing rather than non-decreasing). Other
5719:
5699:
5676:
5610:
5547:
5515:
5509:
is a continuous non-decreasing function whose domain
5473:
5427:
5250:
5228:
5135:
5106:
5082:
5029:
5005:
4921:
4885:
4768:
4742:
4722:
4702:
4682:
4636:
4584:
4552:
4484:
4461:
4415:
4391:
4330:
4310:
4264:
4209:
4189:
4143:
4117:
4068:
4035:
4009:
3989:
3960:
3940:
3917:
3875:
3842:
3816:
3796:
3767:
3747:
3724:
3704:
3662:
3630:
3598:
3572:
3520:
3485:
3459:
3439:
3393:
3369:
3330:
3310:
3290:
3266:
3238:
3223:
3199:
3170:
3126:
3100:
3077:
3057:
3034:
3014:
2970:
2950:
2930:
2900:
2877:
2857:
2832:
2811:
2791:
2763:
2741:
2721:
2698:
2678:
2645:
2625:
2605:
2580:
2546:
2526:
2503:
2483:
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2420:
2400:
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2336:
2314:
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2174:
2145:
2123:
2091:
2069:
2046:
2026:
1980:
1948:
1925:
1899:
1872:
1852:
1829:
1802:
1779:
1759:
1731:
1711:
1691:
1671:
1651:
1603:
1581:
1528:
1508:
1488:
1468:
1439:
1413:
1391:
1371:
1351:
1327:
1293:
1273:
1241:
1215:
1192:
1172:
1140:
1120:
1092:
1066:
1044:
1024:
1004:
980:
946:
926:
894:
868:
845:
825:
793:
770:
726:
694:
665:
645:
616:
543:
523:
495:
472:
452:
428:
408:
364:
340:
312:
289:
269:
245:
222:
179:
159:
139:
100:
80:
60:
40:
9702:
3051:
in the function space containing all functions from
576:
to the concept of a supremum. Infima and suprema of
5222:is finite, then for every non-negative real number
2352:of integers has the least-upper-bound property; if
1796:assuming it exists, does not necessarily belong to
1630:subsets have both a supremum and an infimum, and a
10357:
10006:
9965:
9932:
9872:
9843:
9802:
9738:
9692:
9656:
9624:
9589:
9560:
9428:
9406:
9382:
9362:
9317:
9297:
9275:
9244:
9222:
9187:
9165:
9142:
9119:
8958:
8884:
8803:
8702:
8598:
8578:
8558:
8511:
8472:
8450:{\displaystyle \left(x_{n}\right)_{n=1}^{\infty }}
8449:
8394:
8290:
8211:
8149:
8087:
8067:
8047:
7996:
7937:
7887:
7860:
7840:
7820:
7800:
7732:
7697:
7677:
7643:
7580:
7544:
7499:
7466:
7387:
7354:
7321:
7259:
7219:
7148:
7125:
7098:
7017:
6946:
6917:
6818:
6792:
6693:
6665:
6603:
6583:
6556:
6488:
6423:
6403:
6381:
6304:
6284:
6257:
6189:
6124:
6050:
6030:
5998:
5956:
5933:
5910:
5886:
5838:
5804:
5769:
5749:
5725:
5705:
5685:
5660:
5596:
5533:
5501:
5459:
5413:
5237:
5214:
5118:
5088:
5068:
5011:
4991:
4903:
4871:
4754:
4728:
4708:
4688:
4668:
4622:
4570:
4538:
4470:
4447:
4397:
4371:
4316:
4296:
4250:
4195:
4175:
4129:
4096:
4054:
4021:
3995:
3975:
3946:
3926:
3903:
3861:
3828:
3802:
3782:
3753:
3733:
3710:
3683:
3648:
3616:
3584:
3556:
3506:
3471:
3445:
3418:
3375:
3355:
3316:
3296:
3272:
3244:
3205:
3185:
3156:
3112:
3086:
3063:
3043:
3020:
2986:
2956:
2936:
2916:
2886:
2863:
2842:
2817:
2797:
2777:
2749:
2727:
2707:
2684:
2664:
2631:
2611:
2591:
2555:
2532:
2512:
2489:
2469:
2446:
2426:
2406:
2386:
2364:
2344:
2322:
2300:
2280:
2260:
2211:
2189:
2160:
2131:
2109:
2077:
2052:
2032:
1986:
1966:
1934:
1908:
1881:
1858:
1838:
1811:
1788:
1765:
1737:
1717:
1697:
1677:
1657:
1611:
1589:
1567:
1514:
1494:
1474:
1445:
1425:
1400:
1377:
1357:
1333:
1299:
1279:
1256:
1227:
1198:
1178:
1158:
1126:
1098:
1078:
1053:
1030:
1010:
986:
952:
932:
909:
880:
851:
831:
811:
776:
735:
712:
680:
651:
631:
595:The concepts of infimum and supremum are close to
549:
529:
501:
481:
458:
434:
414:
370:
346:
318:
298:
275:
251:
228:
185:
165:
145:
109:
86:
66:
46:
5713:with appropriate changes (such as requiring that
4372:{\displaystyle \lim _{n\to \infty }s_{n}=\inf S.}
4251:{\displaystyle \lim _{n\to \infty }s_{n}=\sup S.}
466:that is greater than or equal to each element of
10667:
10133:Breitenbach, Jerome R. & Weisstein, Eric W.
9948:
9912:
9895:
9782:
9765:
9645:
9610:
9084:
9048:
8991:
8900:
8873:
8864:
8819:
8718:
8656:
8615:
8487:
8306:
8227:
8165:
8111:
7917:
7905:
7620:
7603:
7560:
7545:{\displaystyle \sup _{}{\tfrac {1}{S}}=\infty ,}
7515:
7482:
7434:
7409:
7337:
7303:
7281:
7087:
7066:
7053:
7032:
6903:
6876:
6860:
6833:
6778:
6751:
6735:
6708:
6542:
6527:
6503:
6477:
6462:
6438:
6431:are nonempty sets of positive real numbers then
6243:
6228:
6204:
6178:
6163:
6139:
5967:
5948:
5925:
5764:
5744:
5700:
5677:
5611:
5357:
5305:
5175:
5054:
5030:
4892:
4830:
4796:
4775:
4743:
4623:{\displaystyle \lim _{n\to \infty }s_{n}=\sup S}
4614:
4586:
4360:
4332:
4239:
4211:
3967:
3918:
3774:
3725:
3672:
3663:
3608:
3599:
3539:
3530:
3495:
3486:
3395:
3332:
3324:is not bounded below, one often formally writes
1568:{\displaystyle \{x\in \mathbb {Q} :x^{2}<2\}}
10068:
10040:
4716:is a continuous function whose domain contains
2268:has the property that every nonempty subset of
580:are common special cases that are important in
9452: – Infimum and supremum almost everywhere
8969:In the last example, the supremum of a set of
489:if such an element exists. If the supremum of
283:that is less than or equal to each element of
10164:
9750:; in fact, with this definition the equality
5460:{\displaystyle f:[0,\infty )\to \mathbb {R} }
4097:{\displaystyle a_{\epsilon }>p-\epsilon .}
3904:{\displaystyle a_{\epsilon }<p+\epsilon .}
2222:
306:if such an element exists. If the infimum of
9483: – Majorant and minorant in mathematics
9114:
9087:
9078:
9051:
9036:
8994:
8858:
8822:
8691:
8659:
8650:
8618:
8553:
8535:
8200:
8168:
8138:
8114:
8042:
8024:
7988:
7961:
7009:
6988:
6657:
6633:
6373:
6337:
6258:{\displaystyle \sup(A+B)=(\sup A)+(\sup B).}
6125:{\displaystyle A+B~:=~\{a+b:a\in A,b\in B\}}
6119:
6083:
5646:
5639:
5591:
5554:
5258:
5251:
5143:
5136:
5023:, then it is even possible to conclude that
4983:
4956:
1562:
1529:
9997:
9995:
9993:
9991:
9989:
9461: – Element ≥ (or ≤) each other element
6190:{\displaystyle \inf(A+B)=(\inf A)+(\inf B)}
4297:{\displaystyle s_{1}\geq s_{2}\geq \cdots }
4176:{\displaystyle s_{1}\leq s_{2}\leq \cdots }
3280:(which is not a negative real number). The
1457:
10588:Vitale's random Brunn–Minkowski inequality
10171:
10157:
7179:
6666:{\displaystyle rB~:=~\{r\cdot b:b\in B\}.}
5887:{\displaystyle L^{\infty }(\Omega ,\mu ),}
5597:{\displaystyle S:=\{|g(x)|:x\in \Omega \}}
27:Greatest lower bound and least upper bound
9425:
9421:
9241:
9237:
9216:
9206:
8915:
8669:
8628:
8242:
8178:
5999:{\displaystyle A,B\subseteq \mathbb {R} }
5992:
5638:
5617:
5453:
5173:
5152:
4955:
4934:
2771:
2743:
2582:
2380:
2338:
2316:
2205:
2177:
2148:
2125:
2094:
2071:
1605:
1583:
1539:
697:
668:
619:
9986:
9810:will also hold for any non-empty subset
9739:{\displaystyle {\tfrac {1}{\infty }}:=0}
9450:Essential supremum and essential infimum
8150:{\displaystyle \inf\{1,2,3,\ldots \}=1.}
5119:{\displaystyle \Omega \neq \varnothing }
1749:Relation to maximum and minimum elements
1626:is a partially ordered set in which all
751:
358:is less than or equal to the infimum of
128:
29:
9844:{\displaystyle S\subseteq (0,\infty ].}
7888:{\displaystyle P^{\operatorname {op} }}
7678:{\displaystyle P^{\operatorname {op} }}
7388:{\displaystyle {\frac {1}{\infty }}:=0}
5693:similar conclusions can be reached for
3213:-tuples and sequences of real numbers.
2007:
14:
10668:
9223:{\displaystyle (\mathbb {N} ,\mid \,)}
8984:One basic property of the supremum is
7260:{\displaystyle S\subseteq (0,\infty )}
2520:an integer that is an upper bound for
1942:there is another negative real number
1106:is larger than any other lower bound).
10152:
10132:
10082:Springer Science & Business Media
10052:
10016:(3rd ed.). McGraw-Hill. p.
10001:
8977:, which means that the rationals are
7018:{\textstyle -A:=(-1)A=\{-a:a\in A\},}
4879:which (for instance) guarantees that
2085:and the set of positive real numbers
94:(filled circles), and the infimum of
10601:Applications & related
10046:
8457:is a decreasing sequence with limit
7997:{\displaystyle -S:=\{-s~:~s\in S\}.}
7329:where this equation also holds when
5670:Although this discussion focused on
5069:{\displaystyle \sup f(S)=f(\sup S).}
3649:{\displaystyle A=\varnothing \neq B}
2237:is an example of the aforementioned
1634:is a partially ordered set in which
1453:is less than any other upper bound).
747:
572:The infimum is, in a precise sense,
10520:Marcinkiewicz interpolation theorem
10009:Principles of Mathematical Analysis
8528:The supremum of the set of numbers
4669:{\displaystyle s_{1},s_{2},\ldots }
4448:{\displaystyle s_{1},s_{2},\ldots }
4003:is an upper bound and if for every
3419:{\displaystyle \inf _{}S=+\infty .}
3356:{\displaystyle \inf _{}S=-\infty .}
2619:be the set of all rational numbers
395:
206:
24:
10446:Symmetric decreasing rearrangement
10350:
9832:
9724:
9459:Greatest element and least element
8442:
8017:The infimum of the set of numbers
7536:
7374:
7349:
7251:
5869:
5861:
5839:{\displaystyle 1\leq p<\infty }
5833:
5650:
5588:
5525:
5443:
5367:
5315:
5262:
5185:
5147:
5107:
4840:
4806:
4596:
4342:
4221:
4130:{\displaystyle S\neq \varnothing }
4055:{\displaystyle a_{\epsilon }\in A}
3862:{\displaystyle a_{\epsilon }\in A}
3684:{\displaystyle \sup A\leq \sup B.}
3548:
3524:
3507:{\displaystyle \sup A\geq \inf A,}
3472:{\displaystyle A\neq \varnothing }
3410:
3347:
3224:Infima and suprema of real numbers
2599:the set of rational numbers. Let
2574:the least-upper-bound property is
2477:then there is a least upper bound
25:
10692:
10107:
9471:Limit superior and limit inferior
9363:{\displaystyle (P(X),\subseteq )}
7938:{\displaystyle \inf S=-\sup(-S),}
7733:{\displaystyle x{\text{ and }}y,}
7355:{\displaystyle \sup _{}S=\infty }
5113:
4124:
3637:
3617:{\displaystyle \inf A\geq \inf B}
3542:
3533:
3466:
2778:{\displaystyle p\in \mathbb {Q} }
2110:{\displaystyle \mathbb {R} ^{+},}
713:{\displaystyle \mathbb {R} ^{+}.}
121:finite sets, the infimum and the
10059:. Trillia Group. pp. 39–42.
3453:is any set of real numbers then
3282:completeness of the real numbers
3232:, infima and suprema of subsets
3220:is an indicator of the suprema.
2987:{\displaystyle p<{\sqrt {2}}}
2917:{\displaystyle p>{\sqrt {2}}}
2190:{\displaystyle \mathbb {R} ^{+}}
2161:{\displaystyle \mathbb {R} ^{+}}
1967:{\displaystyle {\tfrac {x}{2}},}
681:{\displaystyle \mathbb {R} ^{+}}
632:{\displaystyle \mathbb {R} ^{+}}
9966:{\displaystyle \inf _{}S>0.}
9873:{\displaystyle {\tfrac {1}{0}}}
9305:containing subsets of some set
7777:
7771:
7581:{\displaystyle \inf _{}S>0,}
7109:Multiplicative inverse of a set
7065:
7059:
6875:
6869:
6750:
6744:
6272:The multiplication of two sets
4107:Relation to limits of sequences
3810:is a lower bound and for every
9835:
9823:
9684:
9678:
9651:
9642:
9616:
9607:
9584:
9578:
9494:
9429:{\displaystyle \,\subseteq \,}
9357:
9348:
9342:
9336:
9217:
9202:
9099:
9093:
9063:
9057:
9021:
9015:
9006:
9000:
8736:
8726:
8324:
8314:
7929:
7920:
7254:
7242:
7078:
7069:
7044:
7035:
6979:
6970:
6909:
6900:
6891:
6879:
6866:
6857:
6848:
6836:
6784:
6775:
6766:
6754:
6741:
6732:
6723:
6711:
6548:
6539:
6533:
6524:
6518:
6506:
6483:
6474:
6468:
6459:
6453:
6441:
6249:
6240:
6234:
6225:
6219:
6207:
6184:
6175:
6169:
6160:
6154:
6142:
5878:
5866:
5575:
5571:
5565:
5558:
5528:
5516:
5483:
5477:
5449:
5446:
5434:
5396:
5391:
5385:
5378:
5338:
5334:
5328:
5321:
5208:
5204:
5198:
5191:
5100:) valued function with domain
5060:
5051:
5042:
5036:
4968:
4962:
4931:
4925:
4898:
4889:
4837:
4803:
4781:
4772:
4593:
4562:
4556:
4339:
4218:
4022:{\displaystyle \epsilon >0}
3829:{\displaystyle \epsilon >0}
3694:Identifying infima and suprema
3592:are sets of real numbers then
3151:
3145:
3136:
3130:
3008:If in a partially ordered set
2735:) but no least upper bound in
1846:Similarly, if the supremum of
1153:
1141:
806:
794:
13:
1:
10416:Convergence almost everywhere
10178:
9979:
8512:{\displaystyle \inf x_{n}=x.}
6571:The product of a real number
5968:Arithmetic operations on sets
3428:
3157:{\displaystyle f(x)\leq g(x)}
3003:greatest-lower-bound property
2871:) there exists some rational
2592:{\displaystyle \mathbb {Q} ,}
509:exists, it is unique, and if
386:) is also commonly used. The
326:exists, it is unique, and if
9473: – Bounds of a sequence
9465:Maximal and minimal elements
6058:of real numbers is the set
4630:is a real number (where all
3585:{\displaystyle A\subseteq B}
2750:{\displaystyle \mathbb {Q} }
2387:{\displaystyle \mathbb {Z} }
2345:{\displaystyle \mathbb {Z} }
2323:{\displaystyle \mathbb {R} }
2212:{\displaystyle \mathbb {Z} }
2132:{\displaystyle \mathbb {Z} }
2078:{\displaystyle \mathbb {Z} }
1999:maximal and minimal elements
1645:If the supremum of a subset
1612:{\displaystyle \mathbb {Q} }
1590:{\displaystyle \mathbb {Q} }
756:supremum = least upper bound
7:
10583:Prékopa–Leindler inequality
10436:Locally integrable function
10358:{\displaystyle L^{\infty }}
10120:Encyclopedia of Mathematics
10041:Rockafellar & Wets 2009
9443:
8006:
7500:{\displaystyle \inf _{}S=0}
6611:of real numbers is the set
6006:are sets of real numbers.
5534:{\displaystyle [0,\infty )}
2843:{\displaystyle {\sqrt {2}}}
2665:{\displaystyle q^{2}<2.}
1997:However, the definition of
1891:maximum or greatest element
1773:of a partially ordered set
1134:of a partially ordered set
607:. For instance, the set of
422:of a partially ordered set
10:
10697:
10329:Square-integrable function
9746:is commonly used with the
9625:{\displaystyle f(\sup S),}
8522:
7774: if and only if
7685:the partially-ordered set
7653:
6496:and similarly for suprema
5502:{\displaystyle f(x)=x^{q}}
3218:least-upper-bound property
2994:). Another example is the
2235:least-upper-bound property
2229:Least-upper-bound property
2226:
2223:Least-upper-bound property
1665:exists, it is unique. If
10600:
10578:Minkowski–Steiner formula
10548:
10510:
10454:
10403:
10337:
10281:
10250:
10186:
9657:{\displaystyle f(\sup S)}
9245:{\displaystyle \,\mid \,}
9175:The supremum of a subset
8559:{\displaystyle \{1,2,3\}}
8048:{\displaystyle \{2,3,4\}}
8011:
7808:then infimum of a subset
5019:is also an increasing or
4904:{\displaystyle f(\sup S)}
4546:necessarily converges to
3741:is a real number) and if
3001:There is a corresponding
2570:An example of a set that
2454:is less than or equal to
2394:and there is some number
2060:together with the set of
1159:{\displaystyle (P,\leq )}
812:{\displaystyle (P,\leq )}
557:is less than or equal to
378:. Consequently, the term
10561:Isoperimetric inequality
10115:"Upper and lower bounds"
9487:
5898:. Monotone sequences in
3976:{\displaystyle p=\sup A}
3954:is any real number then
3934:is a real number and if
3783:{\displaystyle p=\inf A}
3761:is any real number then
2414:such that every element
2372:is a nonempty subset of
1821:minimum or least element
1753:The infimum of a subset
1458:Existence and uniqueness
10566:Brunn–Minkowski theorem
10070:Rockafellar, R. Tyrrell
10056:Mathematical Analysis I
9940:is given only for when
7848:equals the supremum of
7707:opposite order relation
6819:{\displaystyle r\leq 0}
6694:{\displaystyle r\geq 0}
6567:Scalar product of a set
5805:{\displaystyle L^{p,w}}
5021:non-decreasing function
4755:{\displaystyle \sup S,}
3186:{\displaystyle x\in X.}
3113:{\displaystyle f\leq g}
2239:completeness properties
1640:completeness properties
1426:{\displaystyle z\geq b}
1257:{\displaystyle x\in S.}
1228:{\displaystyle b\geq x}
1079:{\displaystyle y\leq a}
910:{\displaystyle x\in S.}
881:{\displaystyle a\leq x}
537:, then the supremum of
10421:Convergence in measure
10359:
10135:"Infimum and supremum"
9967:
9934:
9874:
9851:However, the notation
9845:
9804:
9740:
9694:
9658:
9626:
9591:
9562:
9481:Upper and lower bounds
9430:
9408:
9384:
9364:
9319:
9299:
9285:The supremum of a set
9277:
9258:lowest common multiple
9246:
9224:
9189:
9167:
9144:
9121:
8960:
8886:
8805:
8704:
8600:
8580:
8560:
8513:
8474:
8451:
8396:
8292:
8213:
8151:
8089:
8069:
8049:
7998:
7939:
7889:
7862:
7842:
7822:
7802:
7734:
7699:
7679:
7645:
7582:
7546:
7501:
7468:
7389:
7356:
7323:
7261:
7221:
7150:
7133:that does not contain
7127:
7100:
7019:
6948:
6919:
6820:
6794:
6695:
6667:
6605:
6585:
6558:
6490:
6425:
6405:
6383:
6306:
6286:
6259:
6191:
6126:
6052:
6032:
6000:
5958:
5957:{\displaystyle \inf S}
5935:
5934:{\displaystyle \sup S}
5912:
5888:
5840:
5806:
5771:
5751:
5727:
5707:
5687:
5686:{\displaystyle \sup ,}
5662:
5598:
5535:
5503:
5461:
5415:
5239:
5216:
5120:
5090:
5070:
5013:
4993:
4905:
4873:
4756:
4730:
4710:
4690:
4670:
4624:
4572:
4540:
4472:
4449:
4399:
4373:
4318:
4298:
4252:
4197:
4177:
4131:
4098:
4056:
4023:
3997:
3977:
3948:
3928:
3927:{\displaystyle \sup A}
3905:
3863:
3830:
3804:
3784:
3755:
3735:
3734:{\displaystyle \inf A}
3712:
3685:
3650:
3618:
3586:
3558:
3508:
3473:
3447:
3420:
3377:
3357:
3318:
3298:
3274:
3246:
3207:
3187:
3158:
3114:
3088:
3065:
3045:
3022:
2988:
2958:
2938:
2918:
2888:
2865:
2844:
2819:
2799:
2779:
2751:
2729:
2709:
2686:
2666:
2633:
2613:
2593:
2557:
2534:
2514:
2491:
2471:
2448:
2428:
2408:
2388:
2366:
2346:
2324:
2302:
2282:
2262:
2213:
2191:
2162:
2133:
2111:
2079:
2054:
2034:
1988:
1968:
1936:
1910:
1883:
1860:
1840:
1813:
1790:
1767:
1739:
1719:
1699:
1679:
1659:
1613:
1591:
1569:
1516:
1496:
1476:
1447:
1427:
1402:
1379:
1359:
1335:
1301:
1281:
1258:
1229:
1200:
1180:
1160:
1128:
1100:
1080:
1055:
1032:
1012:
988:
954:
934:
911:
882:
853:
833:
813:
778:
757:
737:
714:
682:
653:
633:
551:
531:
503:
483:
460:
436:
416:
372:
348:
320:
300:
277:
253:
230:
194:
187:
167:
147:
126:
111:
88:
68:
48:
10535:Riesz–Fischer theorem
10360:
10319:Polarization identity
10053:Zakon, Elias (2004).
9968:
9935:
9875:
9846:
9805:
9748:extended real numbers
9741:
9695:
9693:{\displaystyle f(S).}
9659:
9632:this guarantees that
9627:
9592:
9563:
9431:
9409:
9385:
9365:
9320:
9300:
9278:
9247:
9225:
9190:
9168:
9145:
9122:
8961:
8887:
8806:
8705:
8601:
8581:
8561:
8514:
8475:
8452:
8397:
8293:
8214:
8152:
8090:
8070:
8050:
7999:
7940:
7890:
7863:
7843:
7823:
7803:
7735:
7700:
7680:
7646:
7583:
7547:
7502:
7469:
7390:
7357:
7324:
7262:
7222:
7151:
7128:
7101:
7020:
6949:
6920:
6821:
6795:
6696:
6668:
6606:
6586:
6559:
6491:
6426:
6406:
6384:
6307:
6287:
6260:
6192:
6127:
6053:
6033:
6001:
5959:
5936:
5913:
5889:
5841:
5807:
5772:
5770:{\displaystyle \inf }
5752:
5750:{\displaystyle \sup }
5728:
5708:
5706:{\displaystyle \inf }
5688:
5663:
5599:
5536:
5504:
5462:
5416:
5240:
5217:
5121:
5091:
5071:
5014:
4994:
4906:
4874:
4757:
4731:
4711:
4691:
4671:
4625:
4573:
4571:{\displaystyle f(p).}
4541:
4473:
4450:
4400:
4374:
4319:
4299:
4253:
4198:
4178:
4132:
4099:
4057:
4024:
3998:
3978:
3949:
3929:
3906:
3864:
3831:
3805:
3785:
3756:
3736:
3713:
3686:
3651:
3619:
3587:
3559:
3509:
3474:
3448:
3421:
3378:
3358:
3319:
3299:
3275:
3247:
3208:
3188:
3159:
3115:
3089:
3066:
3046:
3023:
2989:
2959:
2939:
2919:
2889:
2866:
2845:
2820:
2800:
2780:
2752:
2730:
2710:
2708:{\displaystyle 1000,}
2687:
2667:
2634:
2614:
2594:
2558:
2535:
2515:
2492:
2472:
2449:
2429:
2409:
2389:
2367:
2347:
2325:
2303:
2283:
2263:
2243:Dedekind completeness
2214:
2192:
2163:
2134:
2112:
2080:
2055:
2035:
1989:
1969:
1937:
1911:
1884:
1861:
1841:
1814:
1791:
1768:
1745:(or does not exist).
1740:
1720:
1700:
1680:
1660:
1614:
1592:
1570:
1517:
1497:
1477:
1448:
1428:
1403:
1380:
1360:
1345:for all upper bounds
1336:
1302:
1282:
1259:
1230:
1201:
1181:
1161:
1129:
1101:
1081:
1056:
1033:
1013:
998:for all lower bounds
989:
955:
935:
912:
883:
854:
834:
814:
786:partially ordered set
779:
755:
738:
715:
683:
654:
634:
609:positive real numbers
605:no minimum or maximum
552:
532:
504:
484:
461:
437:
417:
373:
349:
321:
301:
278:
254:
238:partially ordered set
231:
188:
168:
148:
132:
112:
89:
69:
49:
33:
10540:Riesz–Thorin theorem
10383:Infimum and supremum
10342:
10268:Lebesgue integration
10078:Variational Analysis
9944:
9884:
9855:
9814:
9754:
9715:
9672:
9636:
9601:
9590:{\displaystyle f(S)}
9572:
9504:
9418:
9398:
9374:
9333:
9309:
9289:
9264:
9234:
9199:
9179:
9154:
9134:
8988:
8897:
8816:
8715:
8612:
8590:
8570:
8532:
8484:
8461:
8408:
8303:
8224:
8162:
8108:
8079:
8059:
8021:
7949:
7902:
7872:
7852:
7832:
7812:
7744:
7713:
7689:
7662:
7592:
7556:
7511:
7478:
7399:
7366:
7333:
7271:
7233:
7160:
7137:
7117:
7029:
6958:
6947:{\displaystyle r=-1}
6929:
6830:
6804:
6705:
6679:
6615:
6595:
6575:
6500:
6435:
6415:
6395:
6316:
6296:
6276:
6201:
6136:
6062:
6042:
6022:
5976:
5945:
5922:
5902:
5853:
5818:
5783:
5761:
5741:
5737:defined in terms of
5717:
5697:
5674:
5608:
5545:
5513:
5471:
5425:
5248:
5226:
5133:
5104:
5080:
5027:
5003:
4919:
4883:
4766:
4740:
4720:
4700:
4680:
4634:
4582:
4550:
4482:
4459:
4413:
4389:
4328:
4308:
4262:
4207:
4187:
4141:
4115:
4066:
4033:
4007:
3987:
3958:
3938:
3915:
3873:
3840:
3814:
3794:
3765:
3745:
3722:
3702:
3660:
3628:
3596:
3570:
3518:
3483:
3457:
3437:
3391:
3367:
3328:
3308:
3288:
3264:
3236:
3197:
3168:
3124:
3098:
3075:
3055:
3032:
3012:
2968:
2948:
2928:
2898:
2875:
2855:
2830:
2809:
2789:
2761:
2739:
2719:
2696:
2692:has an upper bound (
2676:
2643:
2623:
2603:
2578:
2544:
2524:
2501:
2481:
2458:
2438:
2418:
2398:
2376:
2356:
2334:
2312:
2292:
2272:
2252:
2201:
2172:
2143:
2121:
2089:
2067:
2044:
2024:
2008:Minimal upper bounds
1978:
1946:
1923:
1897:
1870:
1850:
1827:
1819:If it does, it is a
1800:
1777:
1757:
1729:
1709:
1689:
1669:
1649:
1601:
1579:
1526:
1506:
1486:
1466:
1437:
1411:
1389:
1369:
1349:
1325:
1291:
1271:
1239:
1213:
1190:
1170:
1138:
1118:
1090:
1064:
1042:
1022:
1002:
978:
966:greatest lower bound
944:
924:
892:
866:
843:
823:
791:
768:
724:
692:
663:
643:
614:
586:Lebesgue integration
584:, and especially in
541:
521:
493:
470:
450:
426:
406:
380:greatest lower bound
362:
338:
310:
287:
267:
243:
220:
197:In mathematics, the
177:
157:
137:
98:
78:
58:
38:
10502:Young's convolution
10441:Measurable function
10324:Pythagorean theorem
10314:Parseval's identity
10263:Integrable function
9260:of the elements of
8446:
7709:; that is, for all
7267:is non-empty then
5271:
4578:It implies that if
4407:continuous function
2020:As an example, let
10623:Probability theory
10525:Plancherel theorem
10431:Integral transform
10378:Chebyshev distance
10355:
10304:Euclidean distance
10237:Minkowski distance
9963:
9953:
9930:
9928:
9917:
9906:
9900:
9870:
9868:
9841:
9800:
9798:
9787:
9776:
9770:
9736:
9728:
9690:
9654:
9622:
9597:that converges to
9587:
9558:
9426:
9404:
9380:
9360:
9315:
9295:
9276:{\displaystyle S.}
9273:
9242:
9220:
9185:
9166:{\displaystyle g.}
9163:
9140:
9117:
8956:
8882:
8801:
8758:
8700:
8596:
8579:{\displaystyle 3.}
8576:
8556:
8509:
8473:{\displaystyle x,}
8470:
8447:
8411:
8392:
8346:
8288:
8209:
8147:
8085:
8068:{\displaystyle 2.}
8065:
8045:
7994:
7935:
7885:
7858:
7838:
7818:
7798:
7730:
7695:
7675:
7658:If one denotes by
7641:
7636:
7625:
7614:
7608:
7578:
7565:
7542:
7531:
7520:
7497:
7487:
7464:
7459:
7448:
7427:
7423:
7385:
7362:if the definition
7352:
7342:
7319:
7308:
7286:
7257:
7217:
7195:
7149:{\displaystyle 0,}
7146:
7123:
7096:
7015:
6944:
6915:
6816:
6790:
6691:
6663:
6601:
6581:
6554:
6486:
6421:
6401:
6379:
6302:
6282:
6255:
6187:
6122:
6048:
6028:
5996:
5954:
5931:
5908:
5884:
5836:
5802:
5767:
5747:
5723:
5703:
5683:
5658:
5633:
5594:
5531:
5499:
5457:
5411:
5371:
5319:
5290:
5257:
5238:{\displaystyle q,}
5235:
5212:
5189:
5168:
5116:
5086:
5066:
5009:
4989:
4950:
4901:
4869:
4844:
4810:
4752:
4726:
4706:
4686:
4666:
4620:
4600:
4568:
4536:
4471:{\displaystyle p,}
4468:
4445:
4395:
4369:
4346:
4314:
4294:
4248:
4225:
4193:
4173:
4127:
4094:
4052:
4019:
3993:
3973:
3944:
3924:
3901:
3859:
3826:
3800:
3780:
3751:
3731:
3708:
3698:If the infimum of
3681:
3646:
3614:
3582:
3554:
3504:
3469:
3443:
3416:
3400:
3373:
3353:
3337:
3314:
3294:
3270:
3242:
3203:
3183:
3154:
3110:
3087:{\displaystyle P,}
3084:
3061:
3044:{\displaystyle X,}
3041:
3018:
2984:
2954:
2934:
2914:
2887:{\displaystyle r,}
2884:
2861:
2840:
2815:
2795:
2775:
2747:
2725:
2705:
2682:
2662:
2629:
2609:
2589:
2556:{\displaystyle S.}
2553:
2530:
2513:{\displaystyle S,}
2510:
2487:
2470:{\displaystyle n,}
2467:
2444:
2424:
2404:
2384:
2362:
2342:
2320:
2298:
2278:
2258:
2248:If an ordered set
2209:
2187:
2158:
2129:
2107:
2075:
2050:
2030:
1984:
1964:
1959:
1935:{\displaystyle x,}
1932:
1909:{\displaystyle S.}
1906:
1882:{\displaystyle S,}
1879:
1856:
1839:{\displaystyle S.}
1836:
1812:{\displaystyle S.}
1809:
1789:{\displaystyle P,}
1786:
1763:
1735:
1715:
1695:
1675:
1655:
1609:
1587:
1565:
1512:
1492:
1472:
1443:
1423:
1401:{\displaystyle P,}
1398:
1375:
1355:
1331:
1297:
1277:
1254:
1225:
1196:
1176:
1156:
1124:
1096:
1076:
1054:{\displaystyle P,}
1051:
1028:
1008:
984:
950:
930:
907:
878:
849:
829:
809:
774:
758:
736:{\displaystyle 0,}
733:
710:
678:
649:
629:
547:
527:
499:
482:{\displaystyle S,}
479:
456:
432:
412:
368:
344:
316:
299:{\displaystyle S,}
296:
273:
249:
226:
195:
183:
163:
143:
127:
110:{\displaystyle S.}
107:
84:
64:
44:
10663:
10662:
10596:
10595:
10411:Almost everywhere
10196: &
10074:Wets, Roger J.-B.
9947:
9927:
9911:
9905:
9894:
9867:
9797:
9781:
9775:
9764:
9727:
9568:is a sequence in
9407:{\displaystyle X}
9383:{\displaystyle P}
9318:{\displaystyle X}
9298:{\displaystyle S}
9188:{\displaystyle S}
9143:{\displaystyle f}
9047:
9041:
8951:
8757:
8599:{\displaystyle 4}
8345:
8283:
8088:{\displaystyle 1}
7978:
7972:
7861:{\displaystyle S}
7841:{\displaystyle P}
7821:{\displaystyle S}
7790:
7775:
7759:
7722:
7698:{\displaystyle P}
7635:
7619:
7613:
7602:
7559:
7530:
7514:
7481:
7458:
7433:
7428:
7408:
7377:
7336:
7317:
7302:
7301:
7295:
7291:
7280:
7194:
7175:
7171:
7126:{\displaystyle S}
7063:
6954:and the notation
6873:
6748:
6632:
6626:
6604:{\displaystyle B}
6584:{\displaystyle r}
6424:{\displaystyle B}
6404:{\displaystyle A}
6336:
6330:
6305:{\displaystyle B}
6285:{\displaystyle A}
6082:
6076:
6051:{\displaystyle B}
6031:{\displaystyle A}
5918:that converge to
5911:{\displaystyle S}
5726:{\displaystyle f}
5635:
5631:
5356:
5304:
5297:
5292:
5288:
5274:
5174:
5170:
5166:
5089:{\displaystyle g}
5012:{\displaystyle f}
4952:
4948:
4829:
4795:
4729:{\displaystyle S}
4709:{\displaystyle f}
4689:{\displaystyle S}
4585:
4398:{\displaystyle f}
4331:
4317:{\displaystyle S}
4210:
4196:{\displaystyle S}
3996:{\displaystyle p}
3947:{\displaystyle p}
3803:{\displaystyle p}
3754:{\displaystyle p}
3718:exists (that is,
3711:{\displaystyle A}
3446:{\displaystyle A}
3394:
3376:{\displaystyle S}
3331:
3317:{\displaystyle S}
3297:{\displaystyle S}
3273:{\displaystyle 0}
3245:{\displaystyle S}
3206:{\displaystyle n}
3064:{\displaystyle X}
3021:{\displaystyle P}
2982:
2957:{\displaystyle p}
2937:{\displaystyle S}
2924:) or a member of
2912:
2864:{\displaystyle p}
2838:
2818:{\displaystyle y}
2798:{\displaystyle x}
2728:{\displaystyle 6}
2685:{\displaystyle S}
2632:{\displaystyle q}
2612:{\displaystyle S}
2533:{\displaystyle S}
2490:{\displaystyle u}
2447:{\displaystyle S}
2427:{\displaystyle s}
2407:{\displaystyle n}
2365:{\displaystyle S}
2301:{\displaystyle S}
2281:{\displaystyle S}
2261:{\displaystyle S}
2053:{\displaystyle S}
2033:{\displaystyle S}
1987:{\displaystyle 0}
1958:
1859:{\displaystyle S}
1766:{\displaystyle S}
1738:{\displaystyle S}
1718:{\displaystyle S}
1698:{\displaystyle S}
1678:{\displaystyle S}
1658:{\displaystyle S}
1515:{\displaystyle S}
1495:{\displaystyle P}
1475:{\displaystyle S}
1446:{\displaystyle b}
1378:{\displaystyle S}
1358:{\displaystyle z}
1334:{\displaystyle S}
1313:least upper bound
1300:{\displaystyle S}
1280:{\displaystyle b}
1199:{\displaystyle P}
1179:{\displaystyle b}
1127:{\displaystyle S}
1099:{\displaystyle a}
1031:{\displaystyle S}
1011:{\displaystyle y}
987:{\displaystyle S}
953:{\displaystyle S}
933:{\displaystyle a}
852:{\displaystyle P}
832:{\displaystyle a}
777:{\displaystyle S}
748:Formal definition
652:{\displaystyle 0}
563:least upper bound
550:{\displaystyle S}
530:{\displaystyle S}
502:{\displaystyle S}
459:{\displaystyle P}
435:{\displaystyle P}
415:{\displaystyle S}
371:{\displaystyle S}
347:{\displaystyle S}
319:{\displaystyle S}
276:{\displaystyle P}
252:{\displaystyle P}
229:{\displaystyle S}
186:{\displaystyle A}
166:{\displaystyle A}
146:{\displaystyle A}
87:{\displaystyle P}
67:{\displaystyle S}
47:{\displaystyle P}
16:(Redirected from
10688:
10613:Fourier analysis
10571:Milman's reverse
10554:
10552:Lebesgue measure
10546:
10545:
10530:Riemann–Lebesgue
10373:Bounded function
10364:
10362:
10361:
10356:
10354:
10353:
10273:Taxicab geometry
10228:Measurable space
10173:
10166:
10159:
10150:
10149:
10145:
10144:
10128:
10103:
10076:(26 June 2009).
10061:
10060:
10050:
10044:
10038:
10032:
10031:
10015:
10012:
9999:
9973:
9972:
9970:
9969:
9964:
9952:
9939:
9937:
9936:
9931:
9929:
9920:
9916:
9907:
9904:
9899:
9889:
9879:
9877:
9876:
9871:
9869:
9860:
9850:
9848:
9847:
9842:
9809:
9807:
9806:
9801:
9799:
9790:
9786:
9777:
9774:
9769:
9759:
9745:
9743:
9742:
9737:
9729:
9720:
9709:
9700:
9699:
9697:
9696:
9691:
9663:
9661:
9660:
9655:
9631:
9629:
9628:
9623:
9596:
9594:
9593:
9588:
9567:
9565:
9564:
9559:
9551:
9547:
9546:
9527:
9523:
9522:
9498:
9476:
9455:
9435:
9433:
9432:
9427:
9413:
9411:
9410:
9405:
9389:
9387:
9386:
9381:
9369:
9367:
9366:
9361:
9324:
9322:
9321:
9316:
9304:
9302:
9301:
9296:
9282:
9280:
9279:
9274:
9251:
9249:
9248:
9243:
9229:
9227:
9226:
9221:
9209:
9194:
9192:
9191:
9186:
9172:
9170:
9169:
9164:
9149:
9147:
9146:
9141:
9126:
9124:
9123:
9118:
9045:
9039:
8965:
8963:
8962:
8957:
8952:
8947:
8942:
8938:
8931:
8930:
8918:
8891:
8889:
8888:
8883:
8810:
8808:
8807:
8802:
8794:
8790:
8759:
8750:
8744:
8743:
8709:
8707:
8706:
8701:
8672:
8631:
8605:
8603:
8602:
8597:
8585:
8583:
8582:
8577:
8565:
8563:
8562:
8557:
8518:
8516:
8515:
8510:
8499:
8498:
8479:
8477:
8476:
8471:
8456:
8454:
8453:
8448:
8445:
8440:
8429:
8425:
8424:
8401:
8399:
8398:
8393:
8382:
8378:
8347:
8338:
8332:
8331:
8297:
8295:
8294:
8289:
8284:
8282:
8274:
8269:
8265:
8258:
8257:
8245:
8218:
8216:
8215:
8210:
8181:
8156:
8154:
8153:
8148:
8094:
8092:
8091:
8086:
8074:
8072:
8071:
8066:
8054:
8052:
8051:
8046:
8003:
8001:
8000:
7995:
7976:
7970:
7944:
7942:
7941:
7936:
7895:and vice versa.
7894:
7892:
7891:
7886:
7884:
7883:
7867:
7865:
7864:
7859:
7847:
7845:
7844:
7839:
7827:
7825:
7824:
7819:
7807:
7805:
7804:
7799:
7791:
7788:
7776:
7773:
7770:
7769:
7760:
7757:
7739:
7737:
7736:
7731:
7723:
7720:
7704:
7702:
7701:
7696:
7684:
7682:
7681:
7676:
7674:
7673:
7650:
7648:
7647:
7642:
7637:
7628:
7624:
7615:
7612:
7607:
7597:
7587:
7585:
7584:
7579:
7564:
7551:
7549:
7548:
7543:
7532:
7523:
7519:
7506:
7504:
7503:
7498:
7486:
7473:
7471:
7470:
7465:
7460:
7451:
7447:
7429:
7422:
7403:
7394:
7392:
7391:
7386:
7378:
7370:
7361:
7359:
7358:
7353:
7341:
7328:
7326:
7325:
7320:
7318:
7310:
7307:
7299:
7293:
7292:
7290:
7285:
7275:
7266:
7264:
7263:
7258:
7226:
7224:
7223:
7218:
7213:
7209:
7196:
7187:
7173:
7172:
7164:
7155:
7153:
7152:
7147:
7132:
7130:
7129:
7124:
7105:
7103:
7102:
7097:
7064:
7061:
7025:it follows that
7024:
7022:
7021:
7016:
6953:
6951:
6950:
6945:
6924:
6922:
6921:
6916:
6874:
6871:
6825:
6823:
6822:
6817:
6799:
6797:
6796:
6791:
6749:
6746:
6700:
6698:
6697:
6692:
6672:
6670:
6669:
6664:
6630:
6624:
6610:
6608:
6607:
6602:
6590:
6588:
6587:
6582:
6563:
6561:
6560:
6555:
6495:
6493:
6492:
6487:
6430:
6428:
6427:
6422:
6410:
6408:
6407:
6402:
6388:
6386:
6385:
6380:
6334:
6328:
6311:
6309:
6308:
6303:
6291:
6289:
6288:
6283:
6264:
6262:
6261:
6256:
6196:
6194:
6193:
6188:
6131:
6129:
6128:
6123:
6080:
6074:
6057:
6055:
6054:
6049:
6037:
6035:
6034:
6029:
6005:
6003:
6002:
5997:
5995:
5963:
5961:
5960:
5955:
5940:
5938:
5937:
5932:
5917:
5915:
5914:
5909:
5893:
5891:
5890:
5885:
5865:
5864:
5845:
5843:
5842:
5837:
5811:
5809:
5808:
5803:
5801:
5800:
5776:
5774:
5773:
5768:
5756:
5754:
5753:
5748:
5732:
5730:
5729:
5724:
5712:
5710:
5709:
5704:
5692:
5690:
5689:
5684:
5667:
5665:
5664:
5659:
5654:
5653:
5637:
5636:
5634:
5632:
5629:
5625:
5620:
5603:
5601:
5600:
5595:
5578:
5561:
5541:always contains
5540:
5538:
5537:
5532:
5508:
5506:
5505:
5500:
5498:
5497:
5466:
5464:
5463:
5458:
5456:
5420:
5418:
5417:
5412:
5410:
5406:
5405:
5404:
5399:
5381:
5370:
5352:
5351:
5346:
5342:
5341:
5324:
5318:
5295:
5294:
5293:
5291:
5289:
5286:
5282:
5277:
5272:
5270:
5265:
5244:
5242:
5241:
5236:
5221:
5219:
5218:
5213:
5211:
5194:
5188:
5172:
5171:
5169:
5167:
5164:
5160:
5155:
5151:
5150:
5125:
5123:
5122:
5117:
5095:
5093:
5092:
5087:
5075:
5073:
5072:
5067:
5018:
5016:
5015:
5010:
4998:
4996:
4995:
4990:
4954:
4953:
4951:
4949:
4946:
4942:
4937:
4910:
4908:
4907:
4902:
4878:
4876:
4875:
4870:
4865:
4861:
4860:
4843:
4825:
4821:
4820:
4819:
4809:
4761:
4759:
4758:
4753:
4735:
4733:
4732:
4727:
4715:
4713:
4712:
4707:
4695:
4693:
4692:
4687:
4675:
4673:
4672:
4667:
4659:
4658:
4646:
4645:
4629:
4627:
4626:
4621:
4610:
4609:
4599:
4577:
4575:
4574:
4569:
4545:
4543:
4542:
4537:
4529:
4525:
4524:
4505:
4501:
4500:
4477:
4475:
4474:
4469:
4454:
4452:
4451:
4446:
4438:
4437:
4425:
4424:
4404:
4402:
4401:
4396:
4378:
4376:
4375:
4370:
4356:
4355:
4345:
4323:
4321:
4320:
4315:
4303:
4301:
4300:
4295:
4287:
4286:
4274:
4273:
4257:
4255:
4254:
4249:
4235:
4234:
4224:
4202:
4200:
4199:
4194:
4182:
4180:
4179:
4174:
4166:
4165:
4153:
4152:
4136:
4134:
4133:
4128:
4103:
4101:
4100:
4095:
4078:
4077:
4061:
4059:
4058:
4053:
4045:
4044:
4028:
4026:
4025:
4020:
4002:
4000:
3999:
3994:
3982:
3980:
3979:
3974:
3953:
3951:
3950:
3945:
3933:
3931:
3930:
3925:
3910:
3908:
3907:
3902:
3885:
3884:
3868:
3866:
3865:
3860:
3852:
3851:
3835:
3833:
3832:
3827:
3809:
3807:
3806:
3801:
3789:
3787:
3786:
3781:
3760:
3758:
3757:
3752:
3740:
3738:
3737:
3732:
3717:
3715:
3714:
3709:
3690:
3688:
3687:
3682:
3655:
3653:
3652:
3647:
3623:
3621:
3620:
3615:
3591:
3589:
3588:
3583:
3563:
3561:
3560:
3555:
3513:
3511:
3510:
3505:
3478:
3476:
3475:
3470:
3452:
3450:
3449:
3444:
3425:
3423:
3422:
3417:
3399:
3382:
3380:
3379:
3374:
3362:
3360:
3359:
3354:
3336:
3323:
3321:
3320:
3315:
3303:
3301:
3300:
3295:
3279:
3277:
3276:
3271:
3251:
3249:
3248:
3243:
3212:
3210:
3209:
3204:
3192:
3190:
3189:
3184:
3163:
3161:
3160:
3155:
3119:
3117:
3116:
3111:
3093:
3091:
3090:
3085:
3070:
3068:
3067:
3062:
3050:
3048:
3047:
3042:
3027:
3025:
3024:
3019:
2993:
2991:
2990:
2985:
2983:
2978:
2963:
2961:
2960:
2955:
2943:
2941:
2940:
2935:
2923:
2921:
2920:
2915:
2913:
2908:
2893:
2891:
2890:
2885:
2870:
2868:
2867:
2862:
2849:
2847:
2846:
2841:
2839:
2834:
2824:
2822:
2821:
2816:
2804:
2802:
2801:
2796:
2784:
2782:
2781:
2776:
2774:
2757:: If we suppose
2756:
2754:
2753:
2748:
2746:
2734:
2732:
2731:
2726:
2715:for example, or
2714:
2712:
2711:
2706:
2691:
2689:
2688:
2683:
2671:
2669:
2668:
2663:
2655:
2654:
2638:
2636:
2635:
2630:
2618:
2616:
2615:
2610:
2598:
2596:
2595:
2590:
2585:
2562:
2560:
2559:
2554:
2539:
2537:
2536:
2531:
2519:
2517:
2516:
2511:
2496:
2494:
2493:
2488:
2476:
2474:
2473:
2468:
2453:
2451:
2450:
2445:
2433:
2431:
2430:
2425:
2413:
2411:
2410:
2405:
2393:
2391:
2390:
2385:
2383:
2371:
2369:
2368:
2363:
2351:
2349:
2348:
2343:
2341:
2329:
2327:
2326:
2321:
2319:
2307:
2305:
2304:
2299:
2287:
2285:
2284:
2279:
2267:
2265:
2264:
2259:
2218:
2216:
2215:
2210:
2208:
2196:
2194:
2193:
2188:
2186:
2185:
2180:
2167:
2165:
2164:
2159:
2157:
2156:
2151:
2138:
2136:
2135:
2130:
2128:
2116:
2114:
2113:
2108:
2103:
2102:
2097:
2084:
2082:
2081:
2076:
2074:
2059:
2057:
2056:
2051:
2039:
2037:
2036:
2031:
1993:
1991:
1990:
1985:
1973:
1971:
1970:
1965:
1960:
1951:
1941:
1939:
1938:
1933:
1915:
1913:
1912:
1907:
1888:
1886:
1885:
1880:
1865:
1863:
1862:
1857:
1845:
1843:
1842:
1837:
1818:
1816:
1815:
1810:
1795:
1793:
1792:
1787:
1772:
1770:
1769:
1764:
1744:
1742:
1741:
1736:
1724:
1722:
1721:
1716:
1704:
1702:
1701:
1696:
1684:
1682:
1681:
1676:
1664:
1662:
1661:
1656:
1632:complete lattice
1618:
1616:
1615:
1610:
1608:
1596:
1594:
1593:
1588:
1586:
1574:
1572:
1571:
1566:
1555:
1554:
1542:
1521:
1519:
1518:
1513:
1501:
1499:
1498:
1493:
1481:
1479:
1478:
1473:
1452:
1450:
1449:
1444:
1432:
1430:
1429:
1424:
1407:
1405:
1404:
1399:
1384:
1382:
1381:
1376:
1364:
1362:
1361:
1356:
1340:
1338:
1337:
1332:
1306:
1304:
1303:
1298:
1286:
1284:
1283:
1278:
1263:
1261:
1260:
1255:
1234:
1232:
1231:
1226:
1205:
1203:
1202:
1197:
1185:
1183:
1182:
1177:
1165:
1163:
1162:
1157:
1133:
1131:
1130:
1125:
1105:
1103:
1102:
1097:
1085:
1083:
1082:
1077:
1060:
1058:
1057:
1052:
1037:
1035:
1034:
1029:
1017:
1015:
1014:
1009:
993:
991:
990:
985:
959:
957:
956:
951:
939:
937:
936:
931:
916:
914:
913:
908:
887:
885:
884:
879:
858:
856:
855:
850:
838:
836:
835:
830:
818:
816:
815:
810:
783:
781:
780:
775:
742:
740:
739:
734:
719:
717:
716:
711:
706:
705:
700:
687:
685:
684:
679:
677:
676:
671:
658:
656:
655:
650:
638:
636:
635:
630:
628:
627:
622:
556:
554:
553:
548:
536:
534:
533:
528:
508:
506:
505:
500:
488:
486:
485:
480:
465:
463:
462:
457:
441:
439:
438:
433:
421:
419:
418:
413:
397:
382:(abbreviated as
377:
375:
374:
369:
353:
351:
350:
345:
325:
323:
322:
317:
305:
303:
302:
297:
282:
280:
279:
274:
261:greatest element
258:
256:
255:
250:
235:
233:
232:
227:
208:
192:
190:
189:
184:
172:
170:
169:
164:
152:
150:
149:
144:
116:
114:
113:
108:
93:
91:
90:
85:
73:
71:
70:
65:
53:
51:
50:
45:
21:
10696:
10695:
10691:
10690:
10689:
10687:
10686:
10685:
10666:
10665:
10664:
10659:
10592:
10549:
10544:
10506:
10482:Hausdorff–Young
10462:Babenko–Beckner
10450:
10399:
10349:
10345:
10343:
10340:
10339:
10333:
10277:
10246:
10242:Sequence spaces
10182:
10177:
10113:
10110:
10092:
10065:
10064:
10051:
10047:
10043:, pp. 1–2.
10039:
10035:
10028:
10013:
10000:
9987:
9982:
9977:
9976:
9951:
9945:
9942:
9941:
9918:
9915:
9898:
9893:
9887:
9885:
9882:
9881:
9858:
9856:
9853:
9852:
9815:
9812:
9811:
9788:
9785:
9768:
9763:
9757:
9755:
9752:
9751:
9718:
9716:
9713:
9712:
9711:The definition
9710:
9703:
9673:
9670:
9669:
9664:belongs to the
9637:
9634:
9633:
9602:
9599:
9598:
9573:
9570:
9569:
9542:
9538:
9534:
9518:
9514:
9510:
9505:
9502:
9501:
9499:
9495:
9490:
9477:(infimum limit)
9474:
9453:
9446:
9419:
9416:
9415:
9399:
9396:
9395:
9375:
9372:
9371:
9334:
9331:
9330:
9310:
9307:
9306:
9290:
9287:
9286:
9265:
9262:
9261:
9235:
9232:
9231:
9205:
9200:
9197:
9196:
9180:
9177:
9176:
9155:
9152:
9151:
9135:
9132:
9131:
8989:
8986:
8985:
8946:
8926:
8922:
8914:
8907:
8903:
8898:
8895:
8894:
8817:
8814:
8813:
8748:
8739:
8735:
8725:
8721:
8716:
8713:
8712:
8668:
8627:
8613:
8610:
8609:
8591:
8588:
8587:
8571:
8568:
8567:
8533:
8530:
8529:
8525:
8494:
8490:
8485:
8482:
8481:
8462:
8459:
8458:
8441:
8430:
8420:
8416:
8412:
8409:
8406:
8405:
8336:
8327:
8323:
8313:
8309:
8304:
8301:
8300:
8278:
8273:
8253:
8249:
8241:
8234:
8230:
8225:
8222:
8221:
8177:
8163:
8160:
8159:
8109:
8106:
8105:
8080:
8077:
8076:
8060:
8057:
8056:
8022:
8019:
8018:
8014:
8009:
7950:
7947:
7946:
7903:
7900:
7899:
7879:
7875:
7873:
7870:
7869:
7853:
7850:
7849:
7833:
7830:
7829:
7813:
7810:
7809:
7787:
7772:
7765:
7761:
7756:
7745:
7742:
7741:
7721: and
7719:
7714:
7711:
7710:
7690:
7687:
7686:
7669:
7665:
7663:
7660:
7659:
7656:
7626:
7623:
7606:
7601:
7595:
7593:
7590:
7589:
7563:
7557:
7554:
7553:
7521:
7518:
7512:
7509:
7508:
7507:if and only if
7485:
7479:
7476:
7475:
7449:
7437:
7412:
7402:
7400:
7397:
7396:
7369:
7367:
7364:
7363:
7340:
7334:
7331:
7330:
7309:
7306:
7284:
7279:
7274:
7272:
7269:
7268:
7234:
7231:
7230:
7185:
7184:
7180:
7163:
7161:
7158:
7157:
7138:
7135:
7134:
7118:
7115:
7114:
7062: and
7060:
7030:
7027:
7026:
6959:
6956:
6955:
6930:
6927:
6926:
6872: and
6870:
6831:
6828:
6827:
6805:
6802:
6801:
6747: and
6745:
6706:
6703:
6702:
6680:
6677:
6676:
6616:
6613:
6612:
6596:
6593:
6592:
6576:
6573:
6572:
6501:
6498:
6497:
6436:
6433:
6432:
6416:
6413:
6412:
6396:
6393:
6392:
6317:
6314:
6313:
6297:
6294:
6293:
6277:
6274:
6273:
6268:Product of sets
6202:
6199:
6198:
6137:
6134:
6133:
6063:
6060:
6059:
6043:
6040:
6039:
6023:
6020:
6019:
5991:
5977:
5974:
5973:
5970:
5946:
5943:
5942:
5923:
5920:
5919:
5903:
5900:
5899:
5860:
5856:
5854:
5851:
5850:
5846:), the norm on
5819:
5816:
5815:
5790:
5786:
5784:
5781:
5780:
5762:
5759:
5758:
5742:
5739:
5738:
5718:
5715:
5714:
5698:
5695:
5694:
5675:
5672:
5671:
5649:
5645:
5628:
5626:
5621:
5619:
5618:
5609:
5606:
5605:
5574:
5557:
5546:
5543:
5542:
5514:
5511:
5510:
5493:
5489:
5472:
5469:
5468:
5452:
5426:
5423:
5422:
5400:
5395:
5394:
5377:
5376:
5372:
5360:
5347:
5337:
5320:
5308:
5303:
5299:
5298:
5285:
5283:
5278:
5276:
5275:
5266:
5261:
5249:
5246:
5245:
5227:
5224:
5223:
5207:
5190:
5178:
5163:
5161:
5156:
5154:
5153:
5146:
5142:
5134:
5131:
5130:
5105:
5102:
5101:
5081:
5078:
5077:
5028:
5025:
5024:
5004:
5001:
5000:
4945:
4943:
4938:
4936:
4935:
4920:
4917:
4916:
4884:
4881:
4880:
4856:
4852:
4848:
4833:
4815:
4811:
4799:
4794:
4790:
4767:
4764:
4763:
4741:
4738:
4737:
4721:
4718:
4717:
4701:
4698:
4697:
4681:
4678:
4677:
4654:
4650:
4641:
4637:
4635:
4632:
4631:
4605:
4601:
4589:
4583:
4580:
4579:
4551:
4548:
4547:
4520:
4516:
4512:
4496:
4492:
4488:
4483:
4480:
4479:
4460:
4457:
4456:
4433:
4429:
4420:
4416:
4414:
4411:
4410:
4390:
4387:
4386:
4351:
4347:
4335:
4329:
4326:
4325:
4309:
4306:
4305:
4282:
4278:
4269:
4265:
4263:
4260:
4259:
4230:
4226:
4214:
4208:
4205:
4204:
4188:
4185:
4184:
4161:
4157:
4148:
4144:
4142:
4139:
4138:
4116:
4113:
4112:
4073:
4069:
4067:
4064:
4063:
4040:
4036:
4034:
4031:
4030:
4008:
4005:
4004:
3988:
3985:
3984:
3983:if and only if
3959:
3956:
3955:
3939:
3936:
3935:
3916:
3913:
3912:
3880:
3876:
3874:
3871:
3870:
3847:
3843:
3841:
3838:
3837:
3815:
3812:
3811:
3795:
3792:
3791:
3790:if and only if
3766:
3763:
3762:
3746:
3743:
3742:
3723:
3720:
3719:
3703:
3700:
3699:
3661:
3658:
3657:
3629:
3626:
3625:
3597:
3594:
3593:
3571:
3568:
3567:
3519:
3516:
3515:
3484:
3481:
3480:
3479:if and only if
3458:
3455:
3454:
3438:
3435:
3434:
3431:
3398:
3392:
3389:
3388:
3368:
3365:
3364:
3335:
3329:
3326:
3325:
3309:
3306:
3305:
3289:
3286:
3285:
3265:
3262:
3261:
3237:
3234:
3233:
3226:
3198:
3195:
3194:
3169:
3166:
3165:
3125:
3122:
3121:
3120:if and only if
3099:
3096:
3095:
3076:
3073:
3072:
3056:
3053:
3052:
3033:
3030:
3029:
3013:
3010:
3009:
2977:
2969:
2966:
2965:
2949:
2946:
2945:
2929:
2926:
2925:
2907:
2899:
2896:
2895:
2876:
2873:
2872:
2856:
2853:
2852:
2833:
2831:
2828:
2827:
2810:
2807:
2806:
2790:
2787:
2786:
2770:
2762:
2759:
2758:
2742:
2740:
2737:
2736:
2720:
2717:
2716:
2697:
2694:
2693:
2677:
2674:
2673:
2650:
2646:
2644:
2641:
2640:
2624:
2621:
2620:
2604:
2601:
2600:
2581:
2579:
2576:
2575:
2545:
2542:
2541:
2525:
2522:
2521:
2502:
2499:
2498:
2482:
2479:
2478:
2459:
2456:
2455:
2439:
2436:
2435:
2419:
2416:
2415:
2399:
2396:
2395:
2379:
2377:
2374:
2373:
2357:
2354:
2353:
2337:
2335:
2332:
2331:
2315:
2313:
2310:
2309:
2293:
2290:
2289:
2273:
2270:
2269:
2253:
2250:
2249:
2231:
2225:
2204:
2202:
2199:
2198:
2181:
2176:
2175:
2173:
2170:
2169:
2152:
2147:
2146:
2144:
2141:
2140:
2124:
2122:
2119:
2118:
2098:
2093:
2092:
2090:
2087:
2086:
2070:
2068:
2065:
2064:
2045:
2042:
2041:
2025:
2022:
2021:
2010:
1979:
1976:
1975:
1949:
1947:
1944:
1943:
1924:
1921:
1920:
1898:
1895:
1894:
1871:
1868:
1867:
1851:
1848:
1847:
1828:
1825:
1824:
1801:
1798:
1797:
1778:
1775:
1774:
1758:
1755:
1754:
1751:
1730:
1727:
1726:
1710:
1707:
1706:
1690:
1687:
1686:
1670:
1667:
1666:
1650:
1647:
1646:
1628:nonempty finite
1604:
1602:
1599:
1598:
1582:
1580:
1577:
1576:
1550:
1546:
1538:
1527:
1524:
1523:
1507:
1504:
1503:
1487:
1484:
1483:
1467:
1464:
1463:
1460:
1438:
1435:
1434:
1412:
1409:
1408:
1390:
1387:
1386:
1370:
1367:
1366:
1350:
1347:
1346:
1326:
1323:
1322:
1292:
1289:
1288:
1272:
1269:
1268:
1267:An upper bound
1240:
1237:
1236:
1214:
1211:
1210:
1191:
1188:
1187:
1171:
1168:
1167:
1139:
1136:
1135:
1119:
1116:
1115:
1091:
1088:
1087:
1065:
1062:
1061:
1043:
1040:
1039:
1023:
1020:
1019:
1003:
1000:
999:
979:
976:
975:
945:
942:
941:
925:
922:
921:
893:
890:
889:
867:
864:
863:
844:
841:
840:
824:
821:
820:
792:
789:
788:
769:
766:
765:
750:
725:
722:
721:
701:
696:
695:
693:
690:
689:
672:
667:
666:
664:
661:
660:
644:
641:
640:
639:(not including
623:
618:
617:
615:
612:
611:
542:
539:
538:
522:
519:
518:
494:
491:
490:
471:
468:
467:
451:
448:
447:
427:
424:
423:
407:
404:
403:
363:
360:
359:
339:
336:
335:
311:
308:
307:
288:
285:
284:
268:
265:
264:
244:
241:
240:
221:
218:
217:
178:
175:
174:
158:
155:
154:
138:
135:
134:
119:totally ordered
99:
96:
95:
79:
76:
75:
59:
56:
55:
39:
36:
35:
28:
23:
22:
15:
12:
11:
5:
10694:
10684:
10683:
10678:
10661:
10660:
10658:
10657:
10656:
10655:
10650:
10640:
10635:
10630:
10625:
10620:
10615:
10610:
10604:
10602:
10598:
10597:
10594:
10593:
10591:
10590:
10585:
10580:
10575:
10574:
10573:
10563:
10557:
10555:
10543:
10542:
10537:
10532:
10527:
10522:
10516:
10514:
10508:
10507:
10505:
10504:
10499:
10494:
10489:
10484:
10479:
10474:
10469:
10464:
10458:
10456:
10452:
10451:
10449:
10448:
10443:
10438:
10433:
10428:
10426:Function space
10423:
10418:
10413:
10407:
10405:
10401:
10400:
10398:
10397:
10392:
10391:
10390:
10380:
10375:
10369:
10367:
10352:
10348:
10335:
10334:
10332:
10331:
10326:
10321:
10316:
10311:
10306:
10301:
10299:Cauchy–Schwarz
10296:
10290:
10288:
10279:
10278:
10276:
10275:
10270:
10265:
10259:
10257:
10248:
10247:
10245:
10244:
10239:
10234:
10225:
10220:
10219:
10218:
10208:
10200:
10198:Hilbert spaces
10190:
10188:
10187:Basic concepts
10184:
10183:
10176:
10175:
10168:
10161:
10153:
10147:
10146:
10129:
10109:
10108:External links
10106:
10105:
10104:
10090:
10063:
10062:
10045:
10033:
10026:
9984:
9983:
9981:
9978:
9975:
9974:
9962:
9959:
9956:
9950:
9926:
9923:
9914:
9910:
9903:
9897:
9892:
9866:
9863:
9840:
9837:
9834:
9831:
9828:
9825:
9822:
9819:
9796:
9793:
9784:
9780:
9773:
9767:
9762:
9735:
9732:
9726:
9723:
9701:
9689:
9686:
9683:
9680:
9677:
9653:
9650:
9647:
9644:
9641:
9621:
9618:
9615:
9612:
9609:
9606:
9586:
9583:
9580:
9577:
9557:
9554:
9550:
9545:
9541:
9537:
9533:
9530:
9526:
9521:
9517:
9513:
9509:
9492:
9491:
9489:
9486:
9485:
9484:
9478:
9468:
9462:
9456:
9445:
9442:
9424:
9403:
9379:
9359:
9356:
9353:
9350:
9347:
9344:
9341:
9338:
9314:
9294:
9272:
9269:
9240:
9219:
9215:
9212:
9208:
9204:
9184:
9162:
9159:
9139:
9116:
9113:
9110:
9107:
9104:
9101:
9098:
9095:
9092:
9089:
9086:
9083:
9080:
9077:
9074:
9071:
9068:
9065:
9062:
9059:
9056:
9053:
9050:
9044:
9038:
9035:
9032:
9029:
9026:
9023:
9020:
9017:
9014:
9011:
9008:
9005:
9002:
8999:
8996:
8993:
8967:
8966:
8955:
8950:
8945:
8941:
8937:
8934:
8929:
8925:
8921:
8917:
8913:
8910:
8906:
8902:
8892:
8881:
8878:
8875:
8872:
8869:
8866:
8863:
8860:
8857:
8854:
8851:
8848:
8845:
8842:
8839:
8836:
8833:
8830:
8827:
8824:
8821:
8811:
8800:
8797:
8793:
8789:
8786:
8783:
8780:
8777:
8774:
8771:
8768:
8765:
8762:
8756:
8753:
8747:
8742:
8738:
8734:
8731:
8728:
8724:
8720:
8710:
8699:
8696:
8693:
8690:
8687:
8684:
8681:
8678:
8675:
8671:
8667:
8664:
8661:
8658:
8655:
8652:
8649:
8646:
8643:
8640:
8637:
8634:
8630:
8626:
8623:
8620:
8617:
8607:
8595:
8575:
8555:
8552:
8549:
8546:
8543:
8540:
8537:
8524:
8521:
8520:
8519:
8508:
8505:
8502:
8497:
8493:
8489:
8469:
8466:
8444:
8439:
8436:
8433:
8428:
8423:
8419:
8415:
8402:
8391:
8388:
8385:
8381:
8377:
8374:
8371:
8368:
8365:
8362:
8359:
8356:
8353:
8350:
8344:
8341:
8335:
8330:
8326:
8322:
8319:
8316:
8312:
8308:
8298:
8287:
8281:
8277:
8272:
8268:
8264:
8261:
8256:
8252:
8248:
8244:
8240:
8237:
8233:
8229:
8219:
8208:
8205:
8202:
8199:
8196:
8193:
8190:
8187:
8184:
8180:
8176:
8173:
8170:
8167:
8157:
8146:
8143:
8140:
8137:
8134:
8131:
8128:
8125:
8122:
8119:
8116:
8113:
8103:
8096:
8084:
8064:
8044:
8041:
8038:
8035:
8032:
8029:
8026:
8013:
8010:
8008:
8005:
7993:
7990:
7987:
7984:
7981:
7975:
7969:
7966:
7963:
7960:
7957:
7954:
7934:
7931:
7928:
7925:
7922:
7919:
7916:
7913:
7910:
7907:
7882:
7878:
7857:
7837:
7817:
7797:
7794:
7789: in
7786:
7783:
7780:
7768:
7764:
7758: in
7755:
7752:
7749:
7729:
7726:
7718:
7694:
7672:
7668:
7655:
7652:
7640:
7634:
7631:
7622:
7618:
7611:
7605:
7600:
7577:
7574:
7571:
7568:
7562:
7541:
7538:
7535:
7529:
7526:
7517:
7496:
7493:
7490:
7484:
7463:
7457:
7454:
7446:
7443:
7440:
7436:
7432:
7426:
7421:
7418:
7415:
7411:
7406:
7384:
7381:
7376:
7373:
7351:
7348:
7345:
7339:
7316:
7313:
7305:
7298:
7289:
7283:
7278:
7256:
7253:
7250:
7247:
7244:
7241:
7238:
7216:
7212:
7208:
7205:
7202:
7199:
7193:
7190:
7183:
7178:
7170:
7167:
7145:
7142:
7122:
7095:
7092:
7089:
7086:
7083:
7080:
7077:
7074:
7071:
7068:
7058:
7055:
7052:
7049:
7046:
7043:
7040:
7037:
7034:
7014:
7011:
7008:
7005:
7002:
6999:
6996:
6993:
6990:
6987:
6984:
6981:
6978:
6975:
6972:
6969:
6966:
6963:
6943:
6940:
6937:
6934:
6914:
6911:
6908:
6905:
6902:
6899:
6896:
6893:
6890:
6887:
6884:
6881:
6878:
6868:
6865:
6862:
6859:
6856:
6853:
6850:
6847:
6844:
6841:
6838:
6835:
6815:
6812:
6809:
6789:
6786:
6783:
6780:
6777:
6774:
6771:
6768:
6765:
6762:
6759:
6756:
6753:
6743:
6740:
6737:
6734:
6731:
6728:
6725:
6722:
6719:
6716:
6713:
6710:
6690:
6687:
6684:
6662:
6659:
6656:
6653:
6650:
6647:
6644:
6641:
6638:
6635:
6629:
6623:
6620:
6600:
6580:
6553:
6550:
6547:
6544:
6541:
6538:
6535:
6532:
6529:
6526:
6523:
6520:
6517:
6514:
6511:
6508:
6505:
6485:
6482:
6479:
6476:
6473:
6470:
6467:
6464:
6461:
6458:
6455:
6452:
6449:
6446:
6443:
6440:
6420:
6400:
6378:
6375:
6372:
6369:
6366:
6363:
6360:
6357:
6354:
6351:
6348:
6345:
6342:
6339:
6333:
6327:
6324:
6321:
6301:
6281:
6254:
6251:
6248:
6245:
6242:
6239:
6236:
6233:
6230:
6227:
6224:
6221:
6218:
6215:
6212:
6209:
6206:
6186:
6183:
6180:
6177:
6174:
6171:
6168:
6165:
6162:
6159:
6156:
6153:
6150:
6147:
6144:
6141:
6121:
6118:
6115:
6112:
6109:
6106:
6103:
6100:
6097:
6094:
6091:
6088:
6085:
6079:
6073:
6070:
6067:
6047:
6027:
5994:
5990:
5987:
5984:
5981:
5969:
5966:
5953:
5950:
5930:
5927:
5907:
5896:operator norms
5883:
5880:
5877:
5874:
5871:
5868:
5863:
5859:
5848:Lebesgue space
5835:
5832:
5829:
5826:
5823:
5799:
5796:
5793:
5789:
5766:
5746:
5722:
5702:
5682:
5679:
5657:
5652:
5648:
5644:
5641:
5624:
5616:
5613:
5593:
5590:
5587:
5584:
5581:
5577:
5573:
5570:
5567:
5564:
5560:
5556:
5553:
5550:
5530:
5527:
5524:
5521:
5518:
5496:
5492:
5488:
5485:
5482:
5479:
5476:
5455:
5451:
5448:
5445:
5442:
5439:
5436:
5433:
5430:
5421:since the map
5409:
5403:
5398:
5393:
5390:
5387:
5384:
5380:
5375:
5369:
5366:
5363:
5359:
5355:
5350:
5345:
5340:
5336:
5333:
5330:
5327:
5323:
5317:
5314:
5311:
5307:
5302:
5281:
5269:
5264:
5260:
5256:
5253:
5234:
5231:
5210:
5206:
5203:
5200:
5197:
5193:
5187:
5184:
5181:
5177:
5159:
5149:
5145:
5141:
5138:
5115:
5112:
5109:
5096:is a real (or
5085:
5065:
5062:
5059:
5056:
5053:
5050:
5047:
5044:
5041:
5038:
5035:
5032:
5008:
4988:
4985:
4982:
4979:
4976:
4973:
4970:
4967:
4964:
4961:
4958:
4941:
4933:
4930:
4927:
4924:
4913:adherent point
4900:
4897:
4894:
4891:
4888:
4868:
4864:
4859:
4855:
4851:
4847:
4842:
4839:
4836:
4832:
4828:
4824:
4818:
4814:
4808:
4805:
4802:
4798:
4793:
4789:
4786:
4783:
4780:
4777:
4774:
4771:
4751:
4748:
4745:
4725:
4705:
4685:
4665:
4662:
4657:
4653:
4649:
4644:
4640:
4619:
4616:
4613:
4608:
4604:
4598:
4595:
4592:
4588:
4567:
4564:
4561:
4558:
4555:
4535:
4532:
4528:
4523:
4519:
4515:
4511:
4508:
4504:
4499:
4495:
4491:
4487:
4467:
4464:
4444:
4441:
4436:
4432:
4428:
4423:
4419:
4394:
4368:
4365:
4362:
4359:
4354:
4350:
4344:
4341:
4338:
4334:
4313:
4293:
4290:
4285:
4281:
4277:
4272:
4268:
4247:
4244:
4241:
4238:
4233:
4229:
4223:
4220:
4217:
4213:
4192:
4172:
4169:
4164:
4160:
4156:
4151:
4147:
4126:
4123:
4120:
4093:
4090:
4087:
4084:
4081:
4076:
4072:
4051:
4048:
4043:
4039:
4018:
4015:
4012:
3992:
3972:
3969:
3966:
3963:
3943:
3923:
3920:
3911:Similarly, if
3900:
3897:
3894:
3891:
3888:
3883:
3879:
3858:
3855:
3850:
3846:
3825:
3822:
3819:
3799:
3779:
3776:
3773:
3770:
3750:
3730:
3727:
3707:
3680:
3677:
3674:
3671:
3668:
3665:
3645:
3642:
3639:
3636:
3633:
3613:
3610:
3607:
3604:
3601:
3581:
3578:
3575:
3553:
3550:
3547:
3544:
3541:
3538:
3535:
3532:
3529:
3526:
3523:
3514:and otherwise
3503:
3500:
3497:
3494:
3491:
3488:
3468:
3465:
3462:
3442:
3430:
3427:
3415:
3412:
3409:
3406:
3403:
3397:
3372:
3352:
3349:
3346:
3343:
3340:
3334:
3313:
3293:
3269:
3241:
3225:
3222:
3202:
3182:
3179:
3176:
3173:
3153:
3150:
3147:
3144:
3141:
3138:
3135:
3132:
3129:
3109:
3106:
3103:
3083:
3080:
3060:
3040:
3037:
3017:
3004:
2981:
2976:
2973:
2953:
2933:
2911:
2906:
2903:
2883:
2880:
2860:
2837:
2814:
2794:
2773:
2769:
2766:
2745:
2724:
2704:
2701:
2681:
2661:
2658:
2653:
2649:
2628:
2608:
2588:
2584:
2573:
2552:
2549:
2529:
2509:
2506:
2486:
2466:
2463:
2443:
2423:
2403:
2382:
2361:
2340:
2318:
2297:
2277:
2257:
2244:
2236:
2227:Main article:
2224:
2221:
2207:
2184:
2179:
2155:
2150:
2127:
2106:
2101:
2096:
2073:
2049:
2029:
2009:
2006:
1983:
1963:
1957:
1954:
1931:
1928:
1905:
1902:
1878:
1875:
1855:
1835:
1832:
1808:
1805:
1785:
1782:
1762:
1750:
1747:
1734:
1714:
1694:
1674:
1654:
1637:
1629:
1607:
1585:
1564:
1561:
1558:
1553:
1549:
1545:
1541:
1537:
1534:
1531:
1511:
1491:
1471:
1459:
1456:
1455:
1454:
1442:
1422:
1419:
1416:
1397:
1394:
1374:
1354:
1330:
1319:
1314:
1310:
1296:
1276:
1265:
1264:
1253:
1250:
1247:
1244:
1224:
1221:
1218:
1195:
1175:
1166:is an element
1155:
1152:
1149:
1146:
1143:
1123:
1113:
1110:Similarly, an
1108:
1107:
1095:
1075:
1072:
1069:
1050:
1047:
1027:
1007:
983:
972:
967:
963:
949:
929:
920:A lower bound
918:
917:
906:
903:
900:
897:
877:
874:
871:
848:
828:
819:is an element
808:
805:
802:
799:
796:
773:
763:
749:
746:
732:
729:
709:
704:
699:
675:
670:
648:
626:
621:
606:
568:
546:
526:
498:
478:
475:
455:
431:
411:
402:) of a subset
385:
367:
343:
315:
295:
292:
272:
248:
225:
193:(red diamond).
182:
162:
142:
117:Note that for
106:
103:
83:
63:
43:
26:
9:
6:
4:
3:
2:
10693:
10682:
10679:
10677:
10674:
10673:
10671:
10654:
10651:
10649:
10646:
10645:
10644:
10641:
10639:
10638:Sobolev space
10636:
10634:
10633:Real analysis
10631:
10629:
10626:
10624:
10621:
10619:
10618:Lorentz space
10616:
10614:
10611:
10609:
10608:Bochner space
10606:
10605:
10603:
10599:
10589:
10586:
10584:
10581:
10579:
10576:
10572:
10569:
10568:
10567:
10564:
10562:
10559:
10558:
10556:
10553:
10547:
10541:
10538:
10536:
10533:
10531:
10528:
10526:
10523:
10521:
10518:
10517:
10515:
10513:
10509:
10503:
10500:
10498:
10495:
10493:
10490:
10488:
10485:
10483:
10480:
10478:
10475:
10473:
10470:
10468:
10465:
10463:
10460:
10459:
10457:
10453:
10447:
10444:
10442:
10439:
10437:
10434:
10432:
10429:
10427:
10424:
10422:
10419:
10417:
10414:
10412:
10409:
10408:
10406:
10402:
10396:
10393:
10389:
10386:
10385:
10384:
10381:
10379:
10376:
10374:
10371:
10370:
10368:
10366:
10346:
10336:
10330:
10327:
10325:
10322:
10320:
10317:
10315:
10312:
10310:
10309:Hilbert space
10307:
10305:
10302:
10300:
10297:
10295:
10292:
10291:
10289:
10287:
10285:
10280:
10274:
10271:
10269:
10266:
10264:
10261:
10260:
10258:
10256:
10254:
10249:
10243:
10240:
10238:
10235:
10233:
10229:
10226:
10224:
10223:Measure space
10221:
10217:
10214:
10213:
10212:
10209:
10207:
10205:
10201:
10199:
10195:
10192:
10191:
10189:
10185:
10181:
10174:
10169:
10167:
10162:
10160:
10155:
10154:
10151:
10142:
10141:
10136:
10130:
10126:
10122:
10121:
10116:
10112:
10111:
10101:
10097:
10093:
10091:9783642024313
10087:
10083:
10079:
10075:
10071:
10067:
10066:
10058:
10057:
10049:
10042:
10037:
10029:
10027:0-07-054235-X
10023:
10019:
10011:
10010:
10004:
10003:Rudin, Walter
9998:
9996:
9994:
9992:
9990:
9985:
9960:
9957:
9954:
9924:
9921:
9908:
9901:
9890:
9864:
9861:
9838:
9829:
9826:
9820:
9817:
9794:
9791:
9778:
9771:
9760:
9749:
9733:
9730:
9721:
9708:
9706:
9687:
9681:
9675:
9667:
9648:
9639:
9619:
9613:
9604:
9581:
9575:
9555:
9552:
9548:
9543:
9539:
9535:
9531:
9528:
9524:
9519:
9515:
9511:
9507:
9497:
9493:
9482:
9479:
9472:
9469:
9466:
9463:
9460:
9457:
9451:
9448:
9447:
9441:
9439:
9422:
9401:
9393:
9377:
9354:
9351:
9345:
9339:
9328:
9312:
9292:
9283:
9270:
9267:
9259:
9255:
9238:
9213:
9210:
9182:
9173:
9160:
9157:
9137:
9130:
9111:
9108:
9105:
9102:
9096:
9090:
9081:
9075:
9072:
9069:
9066:
9060:
9054:
9042:
9033:
9030:
9027:
9024:
9018:
9012:
9009:
9003:
8997:
8982:
8980:
8976:
8972:
8953:
8948:
8943:
8939:
8935:
8932:
8927:
8923:
8919:
8911:
8908:
8904:
8893:
8879:
8876:
8870:
8867:
8861:
8855:
8852:
8849:
8846:
8843:
8840:
8837:
8834:
8831:
8828:
8825:
8812:
8798:
8795:
8791:
8787:
8784:
8781:
8778:
8775:
8772:
8769:
8766:
8763:
8760:
8754:
8751:
8745:
8740:
8732:
8729:
8722:
8711:
8697:
8694:
8688:
8685:
8682:
8679:
8676:
8673:
8665:
8662:
8653:
8647:
8644:
8641:
8638:
8635:
8632:
8624:
8621:
8608:
8593:
8573:
8550:
8547:
8544:
8541:
8538:
8527:
8526:
8506:
8503:
8500:
8495:
8491:
8467:
8464:
8437:
8434:
8431:
8426:
8421:
8417:
8413:
8403:
8389:
8386:
8383:
8379:
8375:
8372:
8369:
8366:
8363:
8360:
8357:
8354:
8351:
8348:
8342:
8339:
8333:
8328:
8320:
8317:
8310:
8299:
8285:
8279:
8275:
8270:
8266:
8262:
8259:
8254:
8250:
8246:
8238:
8235:
8231:
8220:
8206:
8203:
8197:
8194:
8191:
8188:
8185:
8182:
8174:
8171:
8158:
8144:
8141:
8135:
8132:
8129:
8126:
8123:
8120:
8117:
8104:
8101:
8097:
8082:
8062:
8039:
8036:
8033:
8030:
8027:
8016:
8015:
8004:
7991:
7985:
7982:
7979:
7973:
7967:
7964:
7958:
7955:
7952:
7932:
7926:
7923:
7914:
7911:
7908:
7896:
7880:
7876:
7855:
7835:
7815:
7795:
7792:
7784:
7781:
7778:
7766:
7762:
7753:
7750:
7747:
7727:
7724:
7716:
7708:
7692:
7670:
7666:
7651:
7638:
7632:
7629:
7616:
7609:
7598:
7575:
7572:
7569:
7566:
7539:
7533:
7527:
7524:
7494:
7491:
7488:
7461:
7455:
7452:
7444:
7441:
7438:
7430:
7424:
7419:
7416:
7413:
7404:
7382:
7379:
7371:
7346:
7343:
7314:
7311:
7296:
7287:
7276:
7248:
7245:
7239:
7236:
7227:
7214:
7210:
7206:
7203:
7200:
7197:
7191:
7188:
7181:
7176:
7168:
7165:
7143:
7140:
7120:
7111:
7110:
7106:
7093:
7090:
7084:
7081:
7075:
7072:
7056:
7050:
7047:
7041:
7038:
7012:
7006:
7003:
7000:
6997:
6994:
6991:
6985:
6982:
6976:
6973:
6967:
6964:
6961:
6941:
6938:
6935:
6932:
6912:
6906:
6897:
6894:
6888:
6885:
6882:
6863:
6854:
6851:
6845:
6842:
6839:
6813:
6810:
6807:
6787:
6781:
6772:
6769:
6763:
6760:
6757:
6738:
6729:
6726:
6720:
6717:
6714:
6688:
6685:
6682:
6673:
6660:
6654:
6651:
6648:
6645:
6642:
6639:
6636:
6627:
6621:
6618:
6598:
6578:
6569:
6568:
6564:
6551:
6545:
6536:
6530:
6521:
6515:
6512:
6509:
6480:
6471:
6465:
6456:
6450:
6447:
6444:
6418:
6398:
6389:
6376:
6370:
6367:
6364:
6361:
6358:
6355:
6352:
6349:
6346:
6343:
6340:
6331:
6325:
6322:
6319:
6299:
6279:
6270:
6269:
6265:
6252:
6246:
6237:
6231:
6222:
6216:
6213:
6210:
6181:
6172:
6166:
6157:
6151:
6148:
6145:
6116:
6113:
6110:
6107:
6104:
6101:
6098:
6095:
6092:
6089:
6086:
6077:
6071:
6068:
6065:
6045:
6025:
6017:
6016:Minkowski sum
6012:
6011:
6007:
5988:
5985:
5982:
5979:
5965:
5951:
5928:
5905:
5897:
5881:
5875:
5872:
5857:
5849:
5830:
5827:
5824:
5821:
5813:
5797:
5794:
5791:
5787:
5736:
5720:
5680:
5668:
5655:
5642:
5622:
5614:
5585:
5582:
5579:
5568:
5562:
5551:
5548:
5522:
5519:
5494:
5490:
5486:
5480:
5474:
5440:
5437:
5431:
5428:
5407:
5401:
5388:
5382:
5373:
5364:
5361:
5353:
5348:
5343:
5331:
5325:
5312:
5309:
5300:
5279:
5267:
5254:
5232:
5229:
5201:
5195:
5182:
5179:
5157:
5139:
5129:
5110:
5099:
5083:
5063:
5057:
5048:
5045:
5039:
5033:
5022:
5006:
4986:
4980:
4977:
4974:
4971:
4965:
4959:
4939:
4928:
4922:
4914:
4895:
4886:
4866:
4862:
4857:
4853:
4849:
4845:
4834:
4826:
4822:
4816:
4812:
4800:
4791:
4787:
4784:
4778:
4769:
4749:
4746:
4723:
4703:
4683:
4663:
4660:
4655:
4651:
4647:
4642:
4638:
4617:
4611:
4606:
4602:
4590:
4565:
4559:
4553:
4533:
4530:
4526:
4521:
4517:
4513:
4509:
4506:
4502:
4497:
4493:
4489:
4485:
4465:
4462:
4442:
4439:
4434:
4430:
4426:
4421:
4417:
4408:
4392:
4384:
4379:
4366:
4363:
4357:
4352:
4348:
4336:
4311:
4291:
4288:
4283:
4279:
4275:
4270:
4266:
4245:
4242:
4236:
4231:
4227:
4215:
4190:
4170:
4167:
4162:
4158:
4154:
4149:
4145:
4121:
4118:
4109:
4108:
4104:
4091:
4088:
4085:
4082:
4079:
4074:
4070:
4049:
4046:
4041:
4037:
4016:
4013:
4010:
3990:
3970:
3964:
3961:
3941:
3921:
3898:
3895:
3892:
3889:
3886:
3881:
3877:
3856:
3853:
3848:
3844:
3823:
3820:
3817:
3797:
3777:
3771:
3768:
3748:
3728:
3705:
3696:
3695:
3691:
3678:
3675:
3669:
3666:
3643:
3640:
3634:
3631:
3611:
3605:
3602:
3579:
3576:
3573:
3564:
3551:
3545:
3536:
3527:
3521:
3501:
3498:
3492:
3489:
3463:
3460:
3440:
3426:
3413:
3407:
3404:
3401:
3387:, one writes
3386:
3370:
3350:
3344:
3341:
3338:
3311:
3291:
3283:
3267:
3259:
3255:
3239:
3231:
3221:
3219:
3214:
3200:
3180:
3177:
3174:
3171:
3148:
3142:
3139:
3133:
3127:
3107:
3104:
3101:
3081:
3078:
3058:
3038:
3035:
3015:
3006:
3002:
2999:
2997:
2979:
2974:
2971:
2951:
2944:greater than
2931:
2909:
2904:
2901:
2881:
2878:
2858:
2850:
2835:
2812:
2792:
2767:
2764:
2722:
2702:
2699:
2679:
2659:
2656:
2651:
2647:
2626:
2606:
2586:
2571:
2568:
2566:
2550:
2547:
2527:
2507:
2504:
2484:
2464:
2461:
2441:
2421:
2401:
2359:
2295:
2275:
2255:
2246:
2242:
2240:
2234:
2230:
2220:
2197:smaller than
2182:
2153:
2104:
2099:
2063:
2047:
2027:
2018:
2016:
2005:
2002:
2000:
1995:
1981:
1961:
1955:
1952:
1929:
1926:
1916:
1903:
1900:
1892:
1876:
1873:
1853:
1833:
1830:
1822:
1806:
1803:
1783:
1780:
1760:
1746:
1732:
1712:
1692:
1672:
1652:
1643:
1641:
1635:
1633:
1627:
1625:
1620:
1559:
1556:
1551:
1547:
1543:
1535:
1532:
1509:
1489:
1469:
1440:
1420:
1417:
1414:
1395:
1392:
1372:
1352:
1344:
1343:
1342:
1328:
1320:
1317:
1312:
1308:
1294:
1274:
1251:
1248:
1245:
1242:
1222:
1219:
1216:
1209:
1208:
1207:
1193:
1173:
1150:
1147:
1144:
1121:
1111:
1093:
1073:
1070:
1067:
1048:
1045:
1025:
1005:
997:
996:
995:
981:
973:
970:
965:
961:
960:is called an
947:
927:
904:
901:
898:
895:
875:
872:
869:
862:
861:
860:
846:
826:
803:
800:
797:
787:
771:
761:
754:
745:
730:
727:
707:
702:
673:
646:
624:
610:
604:
602:
598:
593:
591:
587:
583:
579:
575:
570:
566:
564:
560:
544:
524:
516:
512:
496:
476:
473:
453:
445:
444:least element
429:
409:
401:
393:
390:(abbreviated
389:
383:
381:
365:
357:
341:
333:
329:
313:
293:
290:
270:
262:
246:
239:
223:
216:
212:
204:
201:(abbreviated
200:
180:
160:
140:
131:
124:
120:
104:
101:
81:
61:
41:
32:
19:
10681:Superlatives
10676:Order theory
10455:Inequalities
10395:Uniform norm
10382:
10283:
10252:
10203:
10138:
10118:
10077:
10055:
10048:
10036:
10008:
9496:
9284:
9174:
8983:
8968:
7897:
7657:
7228:
7113:For any set
7112:
7108:
7107:
6674:
6570:
6566:
6565:
6390:
6271:
6267:
6266:
6018:of two sets
6013:
6009:
6008:
5971:
5777:include the
5669:
4380:
4110:
4106:
4105:
4029:there is an
3836:there is an
3697:
3693:
3692:
3565:
3432:
3258:real numbers
3254:real numbers
3227:
3215:
3007:
3000:
2569:
2565:well-ordered
2247:
2232:
2019:
2011:
2003:
1996:
1917:
1752:
1644:
1621:
1502:can fail if
1461:
1307:is called a
1266:
1114:of a subset
1109:
919:
764:of a subset
759:
594:
590:order theory
578:real numbers
571:
562:
558:
510:
399:
391:
387:
379:
355:
327:
210:
202:
198:
196:
10653:Von Neumann
10467:Chebyshev's
9129:functionals
8586:The number
8102:of the set.
8075:The number
6010:Sum of sets
5814:norms (for
5467:defined by
4915:of the set
2825:(including
1866:belongs to
1112:upper bound
762:lower bound
515:upper bound
332:lower bound
10670:Categories
10648:C*-algebra
10472:Clarkson's
9980:References
9256:", is the
8979:incomplete
8975:irrational
7740:declare:
7474:Moreover,
6591:and a set
4324:such that
4203:such that
3429:Properties
2996:hyperreals
2639:such that
1206:such that
859:such that
125:are equal.
10643:*-algebra
10628:Quasinorm
10497:Minkowski
10388:Essential
10351:∞
10180:Lp spaces
10140:MathWorld
10125:EMS Press
10100:883392544
9833:∞
9821:⊆
9725:∞
9556:…
9423:⊆
9392:power set
9355:⊆
9252:denotes "
9239:∣
9214:∣
9109:∈
9073:∈
9043:≤
9031:∈
8971:rationals
8912:∈
8853:∈
8841:∈
8788:…
8746:−
8730:−
8686:≤
8680:≤
8666:∈
8625:∈
8443:∞
8387:−
8376:…
8318:−
8239:∈
8175:∈
8136:…
7983:∈
7965:−
7953:−
7924:−
7915:−
7782:≥
7751:≤
7705:with the
7552:where if
7537:∞
7442:∈
7417:∈
7375:∞
7350:∞
7252:∞
7240:⊆
7204:∈
7085:−
7073:−
7051:−
7039:−
7004:∈
6992:−
6974:−
6962:−
6939:−
6886:⋅
6843:⋅
6811:≤
6800:while if
6761:⋅
6718:⋅
6686:≥
6652:∈
6640:⋅
6537:⋅
6513:⋅
6472:⋅
6448:⋅
6368:∈
6356:∈
6344:⋅
6323:⋅
6114:∈
6102:∈
5989:⊆
5876:μ
5870:Ω
5862:∞
5834:∞
5825:≤
5651:∞
5647:‖
5640:‖
5589:Ω
5586:∈
5526:∞
5450:→
5444:∞
5368:Ω
5365:∈
5316:Ω
5313:∈
5263:∞
5259:‖
5252:‖
5186:Ω
5183:∈
5148:∞
5144:‖
5137:‖
5114:∅
5111:≠
5108:Ω
4978:∈
4841:∞
4838:→
4807:∞
4804:→
4696:) and if
4664:…
4597:∞
4594:→
4534:…
4443:…
4343:∞
4340:→
4292:⋯
4289:≥
4276:≥
4222:∞
4219:→
4171:⋯
4168:≤
4155:≤
4125:∅
4122:≠
4089:ϵ
4086:−
4075:ϵ
4047:∈
4042:ϵ
4011:ϵ
3896:ϵ
3882:ϵ
3854:∈
3849:ϵ
3818:ϵ
3670:≤
3641:≠
3638:∅
3606:≥
3577:⊆
3549:∞
3543:∅
3534:∅
3525:∞
3522:−
3493:≥
3467:∅
3464:≠
3411:∞
3348:∞
3345:−
3175:∈
3140:≤
3105:≤
2768:∈
1536:∈
1418:≥
1246:∈
1220:≥
1151:≤
1071:≤
899:∈
873:≤
804:≤
10492:Markov's
10487:Hölder's
10477:Hanner's
10294:Bessel's
10232:function
10216:Lebesgue
9444:See also
9370:, where
9127:for any
8007:Examples
5128:sup norm
4385:that if
4383:topology
3624:(unless
3230:analysis
3164:for all
2062:integers
1889:it is a
1309:supremum
1235:for all
888:for all
582:analysis
388:supremum
10512:Results
10211:Measure
10127:, 2001
10014:(print)
9666:closure
9390:is the
9325:is the
9254:divides
8523:Suprema
8100:minimum
7654:Duality
5941:(or to
5098:complex
4676:are in
3252:of the
1624:lattice
962:infimum
601:maximum
597:minimum
442:is the
400:suprema
354:, then
259:is the
213:) of a
199:infimum
123:minimum
18:Suprema
10365:spaces
10286:spaces
10255:spaces
10206:spaces
10194:Banach
10098:
10088:
10024:
9500:Since
9438:subset
9230:where
9046:
9040:
8012:Infima
7977:
7971:
7945:where
7300:
7294:
7174:
6925:Using
6826:then
6701:then
6631:
6625:
6335:
6329:
6081:
6075:
5296:
5273:
5126:whose
4911:is an
4762:then
3656:) and
3094:where
513:is an
215:subset
211:infima
133:A set
34:A set
9488:Notes
9327:union
8480:then
7588:then
6197:and
5812:space
5779:weak
5735:norms
4478:then
4405:is a
4062:with
3869:with
3385:empty
2672:Then
2572:lacks
2015:total
1321:) of
1315:, or
974:) of
968:, or
784:of a
330:is a
236:of a
10550:For
10404:Maps
10096:OCLC
10086:ISBN
10022:ISBN
9958:>
9414:and
9150:and
8933:<
8645:<
8639:<
8260:>
8195:<
8189:<
7570:>
7156:let
6411:and
6292:and
6038:and
6014:The
5894:and
5831:<
5604:and
4736:and
4409:and
4080:>
4014:>
3887:<
3821:>
3537:<
3216:The
2975:<
2964:(if
2905:>
2851:and
2805:and
2700:1000
2657:<
2497:for
2233:The
2139:and
1557:<
1318:join
1311:(or
971:meet
964:(or
599:and
574:dual
565:(or
9949:inf
9913:sup
9896:inf
9783:inf
9766:sup
9668:of
9646:sup
9611:sup
9436:is
9394:of
9195:of
9085:sup
9049:sup
8992:sup
8973:is
8901:sup
8874:sup
8865:sup
8820:sup
8719:sup
8657:sup
8616:sup
8566:is
8488:inf
8404:If
8307:inf
8228:inf
8166:inf
8112:inf
8055:is
7918:sup
7906:inf
7868:in
7828:in
7621:sup
7604:inf
7561:inf
7516:sup
7483:inf
7435:inf
7410:sup
7338:sup
7304:inf
7282:sup
7229:If
7088:inf
7067:sup
7054:sup
7033:inf
6904:inf
6877:sup
6861:sup
6834:inf
6779:sup
6752:sup
6736:inf
6709:inf
6675:If
6543:sup
6528:sup
6504:sup
6478:inf
6463:inf
6439:inf
6391:If
6244:sup
6229:sup
6205:sup
6179:inf
6164:inf
6140:inf
5949:inf
5926:sup
5765:inf
5757:or
5745:sup
5701:inf
5678:sup
5630:def
5612:sup
5358:sup
5306:sup
5287:def
5176:sup
5165:def
5055:sup
5031:sup
4947:def
4893:sup
4831:lim
4797:lim
4776:sup
4744:sup
4615:sup
4587:lim
4361:inf
4333:lim
4304:in
4240:sup
4212:lim
4183:in
4111:If
3968:sup
3919:sup
3775:inf
3726:inf
3673:sup
3664:sup
3609:inf
3600:inf
3566:If
3540:inf
3531:sup
3496:inf
3487:sup
3433:If
3396:inf
3383:is
3363:If
3333:inf
3228:In
3071:to
2434:of
1893:of
1823:of
1636:all
1619:.)
1575:of
1482:of
1385:in
1365:of
1341:if
1287:of
1186:of
1038:in
1018:of
994:if
940:of
839:of
569:).
567:LUB
517:of
446:in
396:pl.
392:sup
384:GLB
334:of
263:in
207:pl.
203:inf
74:of
10672::
10137:.
10123:,
10117:,
10094:.
10084:.
10072:;
10020:.
9988:^
9961:0.
9731::=
9704:^
9440:.
8981:.
8799:1.
8698:1.
8574:3.
8390:1.
8207:0.
8145:1.
8063:2.
7959::=
7881:op
7767:op
7671:op
7380::=
7177::=
6968::=
6628::=
6332::=
6078::=
5552::=
2660:2.
2563:A
2245:.
1642:.
760:A
398::
394:;
209::
205:;
10347:L
10284:L
10253:L
10230:/
10204:L
10172:e
10165:t
10158:v
10143:.
10102:.
10030:.
10018:4
9955:S
9925:S
9922:1
9909:=
9902:S
9891:1
9865:0
9862:1
9839:.
9836:]
9830:,
9827:0
9824:(
9818:S
9795:S
9792:1
9779:=
9772:S
9761:1
9734:0
9722:1
9688:.
9685:)
9682:S
9679:(
9676:f
9652:)
9649:S
9643:(
9640:f
9620:,
9617:)
9614:S
9608:(
9605:f
9585:)
9582:S
9579:(
9576:f
9553:,
9549:)
9544:2
9540:s
9536:(
9532:f
9529:,
9525:)
9520:1
9516:s
9512:(
9508:f
9402:X
9378:P
9358:)
9352:,
9349:)
9346:X
9343:(
9340:P
9337:(
9313:X
9293:S
9271:.
9268:S
9218:)
9211:,
9207:N
9203:(
9183:S
9161:.
9158:g
9138:f
9115:}
9112:A
9106:t
9103::
9100:)
9097:t
9094:(
9091:g
9088:{
9082:+
9079:}
9076:A
9070:t
9067::
9064:)
9061:t
9058:(
9055:f
9052:{
9037:}
9034:A
9028:t
9025::
9022:)
9019:t
9016:(
9013:g
9010:+
9007:)
9004:t
9001:(
8998:f
8995:{
8954:.
8949:2
8944:=
8940:}
8936:2
8928:2
8924:x
8920::
8916:Q
8909:x
8905:{
8880:.
8877:B
8871:+
8868:A
8862:=
8859:}
8856:B
8850:b
8847:,
8844:A
8838:a
8835::
8832:b
8829:+
8826:a
8823:{
8796:=
8792:}
8785:,
8782:3
8779:,
8776:2
8773:,
8770:1
8767:=
8764:n
8761::
8755:n
8752:1
8741:n
8737:)
8733:1
8727:(
8723:{
8695:=
8692:}
8689:1
8683:x
8677:0
8674::
8670:R
8663:x
8660:{
8654:=
8651:}
8648:1
8642:x
8636:0
8633::
8629:R
8622:x
8619:{
8594:4
8554:}
8551:3
8548:,
8545:2
8542:,
8539:1
8536:{
8507:.
8504:x
8501:=
8496:n
8492:x
8468:,
8465:x
8438:1
8435:=
8432:n
8427:)
8422:n
8418:x
8414:(
8384:=
8380:}
8373:,
8370:3
8367:,
8364:2
8361:,
8358:1
8355:=
8352:n
8349::
8343:n
8340:1
8334:+
8329:n
8325:)
8321:1
8315:(
8311:{
8286:.
8280:3
8276:2
8271:=
8267:}
8263:2
8255:3
8251:x
8247::
8243:Q
8236:x
8232:{
8204:=
8201:}
8198:1
8192:x
8186:0
8183::
8179:R
8172:x
8169:{
8142:=
8139:}
8133:,
8130:3
8127:,
8124:2
8121:,
8118:1
8115:{
8083:1
8043:}
8040:4
8037:,
8034:3
8031:,
8028:2
8025:{
7992:.
7989:}
7986:S
7980:s
7974::
7968:s
7962:{
7956:S
7933:,
7930:)
7927:S
7921:(
7912:=
7909:S
7877:P
7856:S
7836:P
7816:S
7796:,
7793:P
7785:y
7779:x
7763:P
7754:y
7748:x
7728:,
7725:y
7717:x
7693:P
7667:P
7639:.
7633:S
7630:1
7617:=
7610:S
7599:1
7576:,
7573:0
7567:S
7540:,
7534:=
7528:S
7525:1
7495:0
7492:=
7489:S
7462:.
7456:s
7453:1
7445:S
7439:s
7431:=
7425:s
7420:S
7414:s
7405:1
7383:0
7372:1
7347:=
7344:S
7315:S
7312:1
7297:=
7288:S
7277:1
7255:)
7249:,
7246:0
7243:(
7237:S
7215:.
7211:}
7207:S
7201:s
7198::
7192:s
7189:1
7182:{
7169:S
7166:1
7144:,
7141:0
7121:S
7094:.
7091:A
7082:=
7079:)
7076:A
7070:(
7057:A
7048:=
7045:)
7042:A
7036:(
7013:,
7010:}
7007:A
7001:a
6998::
6995:a
6989:{
6986:=
6983:A
6980:)
6977:1
6971:(
6965:A
6942:1
6936:=
6933:r
6913:.
6910:)
6907:A
6901:(
6898:r
6895:=
6892:)
6889:A
6883:r
6880:(
6867:)
6864:A
6858:(
6855:r
6852:=
6849:)
6846:A
6840:r
6837:(
6814:0
6808:r
6788:,
6785:)
6782:A
6776:(
6773:r
6770:=
6767:)
6764:A
6758:r
6755:(
6742:)
6739:A
6733:(
6730:r
6727:=
6724:)
6721:A
6715:r
6712:(
6689:0
6683:r
6661:.
6658:}
6655:B
6649:b
6646::
6643:b
6637:r
6634:{
6622:B
6619:r
6599:B
6579:r
6552:.
6549:)
6546:B
6540:(
6534:)
6531:A
6525:(
6522:=
6519:)
6516:B
6510:A
6507:(
6484:)
6481:B
6475:(
6469:)
6466:A
6460:(
6457:=
6454:)
6451:B
6445:A
6442:(
6419:B
6399:A
6377:.
6374:}
6371:B
6365:b
6362:,
6359:A
6353:a
6350::
6347:b
6341:a
6338:{
6326:B
6320:A
6300:B
6280:A
6253:.
6250:)
6247:B
6241:(
6238:+
6235:)
6232:A
6226:(
6223:=
6220:)
6217:B
6214:+
6211:A
6208:(
6185:)
6182:B
6176:(
6173:+
6170:)
6167:A
6161:(
6158:=
6155:)
6152:B
6149:+
6146:A
6143:(
6120:}
6117:B
6111:b
6108:,
6105:A
6099:a
6096::
6093:b
6090:+
6087:a
6084:{
6072:B
6069:+
6066:A
6046:B
6026:A
5993:R
5986:B
5983:,
5980:A
5952:S
5929:S
5906:S
5882:,
5879:)
5873:,
5867:(
5858:L
5828:p
5822:1
5798:w
5795:,
5792:p
5788:L
5721:f
5681:,
5656:.
5643:g
5623:=
5615:S
5592:}
5583:x
5580::
5576:|
5572:)
5569:x
5566:(
5563:g
5559:|
5555:{
5549:S
5529:)
5523:,
5520:0
5517:[
5495:q
5491:x
5487:=
5484:)
5481:x
5478:(
5475:f
5454:R
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5441:,
5438:0
5435:[
5432::
5429:f
5408:)
5402:q
5397:|
5392:)
5389:x
5386:(
5383:g
5379:|
5374:(
5362:x
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