575:
is compact, a set is uniformly equicontinuous if and only if it is equicontinuous at every point, for essentially the same reason as that uniform continuity and continuity coincide on compact spaces. Used on its own, the term "equicontinuity" may refer to either the pointwise or uniform notion,
5221:
This weaker version is typically used to prove ArzelĂ âAscoli theorem for separable compact spaces. Another consequence is that the limit of an equicontinuous pointwise convergent sequence of continuous functions on a metric space, or on a locally compact space, is continuous. (See below for an
5303:
if it is equicontinuous on the compact set. In practice, showing the equicontinuity is often not so difficult. For example, if the sequence consists of differentiable functions or functions with some regularity (e.g., the functions are solutions of a differential equation), then the
5194:
579:
Some basic properties follow immediately from the definition. Every finite set of continuous functions is equicontinuous. The closure of an equicontinuous set is again equicontinuous. Every member of a uniformly equicontinuous set of functions is
84:) is uniformly convergent if and only if it is equicontinuous and converges pointwise to a function (not necessarily continuous a-priori). In particular, the limit of an equicontinuous pointwise convergent sequence of continuous functions
4907:
4262:
4813:
4658:
3764:
3999:
2795:
538:
3882:
1150:
5291:
This criterion for uniform convergence is often useful in real and complex analysis. Suppose we are given a sequence of continuous functions that converges pointwise on some open subset
1815:
4963:
3183:
4219:
3116:
1016:
4432:
4380:
4314:
2522:
2379:
1946:
1902:
1289:
1687:
1652:
1225:
5308:
or some other kinds of estimates can be used to show the sequence is equicontinuous. It then follows that the limit of the sequence is continuous on every compact subset of
2326:
2263:
3670:
3582:
3020:
2909:
2876:
4184:
4110:
3909:
3791:
3420:
2589:
2632:
70:
4028:
3145:
2408:
2292:
1045:
2978:
2667:
1981:
1713:
1251:
880:
851:
6422:
3367:
3273:
2690:
2191:
1838:
1761:
1577:
1510:
1466:
1398:
4400:
4344:
4282:
4157:
4133:
4068:
4048:
3942:
3833:
3813:
3715:
3691:
3632:
3605:
3552:
3525:
3501:
3474:
3447:
3393:
3344:
3318:
3294:
3250:
3228:
3206:
3060:
3040:
2998:
2949:
2929:
2850:
2815:
2735:
2715:
2544:
2480:
2457:
2433:
2231:
2211:
2168:
2137:
2117:
2093:
2066:
2034:
2010:
1860:
1738:
1617:
1597:
1554:
1531:
1487:
1443:
1421:
1375:
1355:
1331:
1311:
1190:
1170:
1087:
1067:
960:
940:
920:
900:
825:
6071: â sequence of ordinals such that the values assumed at limit stages are the limits (limit suprema and limit infima) of all values at previous stages
5320:
to show the equicontinuity (on a compact subset) and conclude that the limit is holomorphic. Note that the equicontinuity is essential here. For example,
4514:) converges uniformly if and only if it is equicontinuous and converges pointwise. The hypothesis of the statement can be weakened a bit: a sequence in
5373:
of neighbourhoods of one point to be somehow comparable with the filter of neighbourhood of another point. The latter is most generally done via a
6652:
799:(TVS) is a topological group so the definition of an equicontinuous family of maps given for topological groups transfers to TVSs without change.
109:
The uniform boundedness principle states that a pointwise bounded family of continuous linear operators between Banach spaces is equicontinuous.
6196:
Alan F. Beardon, S. Axler, F.W. Gehring, K.A. Ribet : Iteration of
Rational Functions: Complex Analytic Dynamical Systems. Springer, 2000;
597:
is (uniformly) equicontinuous. In particular, this is the case if the set consists of functions with derivatives bounded by the same constant.
6625:
6595:
6531:
6474:
6234:
6209:
6523:
6493:
6406:
6226:
6201:
6115:
6051:
6068:
5876:
The benefit of this generalization is that we may now extend some important definitions that make sense for metric spaces (e.g.
6033:
6657:
4491:
are compact if and only if they are closed and bounded. As a corollary, every uniformly bounded equicontinuous sequence in
6587:
6449:
6088:
6077:
3921:
600:
4846:
4472:
4224:
4736:
4581:
3720:
3947:
6444:
43:
2747:
477:
3838:
1092:
6140:
6023:
5189:{\displaystyle |f_{j}(x)-f_{k}(x)|\leq |f_{j}(x)-f_{j}(z)|+|f_{j}(z)-f_{k}(z)|+|f_{k}(z)-f_{k}(x)|<\epsilon }
1765:
5885:
2853:
1334:
796:
658:
565:
76:, is compact if and only if it is closed, pointwise bounded and equicontinuous. As a corollary, a sequence in
4484:
3152:
4522:) converges uniformly if it is equicontinuous and converges pointwise on a dense subset to some function on
4189:
3065:
965:
4405:
4353:
4287:
6028:
Stochastic equicontinuity is a version of equicontinuity used in the context of sequences of functions of
4483:) is compact if and only if it is closed, uniformly bounded and equicontinuous. This is analogous to the
6144:
5877:
3480:
2879:
2072:
92:
31:
6613:
5846:
needing any metric. Axiomatizing the most basic properties of these sets leads to the definition of a
2485:
2331:
1909:
1865:
1256:
6522:. International Series in Pure and Applied Mathematics. Vol. 8 (Second ed.). New York, NY:
5317:
4082:
2669:
endowed with the topology of bounded convergence (that is, uniform convergence on bounded subsets of
1657:
1622:
1195:
661:
are prominent examples of topological groups and every topological group has an associated canonical
581:
6439:
2299:
2236:
6062:
6045:
5839:
5370:
387:
58:
39:
3645:
3557:
3003:
2884:
2859:
6579:
6567:
6416:
6120:
5835:
5305:
4162:
4088:
3887:
3769:
3635:
3398:
2549:
594:
407:
5316:. A similar argument can be made when the functions are holomorphic. One can use, for instance,
2595:
6631:
6621:
6601:
6591:
6537:
6527:
6499:
6489:
6470:
6402:
6230:
6222:
6205:
6197:
5881:
5374:
5362:
650:
607:
4004:
3121:
2384:
2268:
1021:
6029:
5226: cannot be relaxed. To see that, consider a compactly supported continuous function
3944:
of linear maps between Banach spaces is equicontinuous if it is pointwise bounded; that is,
2957:
2637:
1951:
1692:
1230:
859:
830:
603:
gives a sufficient condition for a set of continuous linear operators to be equicontinuous.
6399:
Asymptotic Theory of
Expanding Parameter Space Methods and Data Dependence in Econometrics
6100:
6094:
6083:
6057:
4347:
4136:
3372:
654:
17:
3349:
3255:
2672:
2173:
1820:
1743:
1559:
1492:
1448:
1380:
6466:
4385:
4329:
4267:
4142:
4118:
4112:
is weak-* compact; thus that every equicontinuous subset is weak-* relatively compact.
4071:
4053:
4033:
3927:
3818:
3798:
3700:
3694:
3676:
3639:
3617:
3608:
3590:
3537:
3510:
3486:
3459:
3432:
3378:
3329:
3303:
3279:
3235:
3213:
3191:
3045:
3025:
2983:
2934:
2914:
2835:
2800:
2720:
2700:
2529:
2465:
2442:
2436:
2418:
2216:
2196:
2153:
2140:
2122:
2102:
2078:
2051:
2037:
2019:
1995:
1845:
1723:
1602:
1582:
1539:
1516:
1472:
1428:
1406:
1360:
1340:
1316:
1296:
1175:
1155:
1072:
1052:
945:
925:
905:
885:
810:
584:, and every finite set of uniformly continuous functions is uniformly equicontinuous.
6646:
6517:
5847:
5666:
5601:
5378:
5345:
2818:
662:
47:
6590:. Vol. 8 (Second ed.). New York, NY: Springer New York Imprint Springer.
6551:
6513:
6080: â Stochastic process that is a continuous function of time or index parameter
4468:
4456:
3528:
3426:
2738:
1989:
235:
if for every ξ > 0, there exists a δ > 0 such that
126:
166:
if for every ξ > 0, there exists a δ > 0 such that
6559:
6458:
6221:
Joseph H. Silverman : The arithmetic of dynamical systems. Springer, 2007.
3453:
2045:
103:
777:
is continuous at the point. Clearly, every finite set of continuous maps from
628:(x) = arctan(nx), is not equicontinuous because the definition is violated at x
4499:) contains a subsequence that converges uniformly to a continuous function on
46:, in a precise sense described herein. In particular, the concept applies to
6635:
6605:
6503:
6488:. Pure and applied mathematics (Second ed.). Boca Raton, FL: CRC Press.
6397:
de Jong, Robert M. (1993). "Stochastic
Equicontinuity for Mixing Processes".
5861:
generate the uniformity that is canonically associated with the metric space
6541:
6106:
4382:
is a compact metrizable space (under the subspace topology). If in addition
3323:
3297:
611:
5299:. As noted above, it actually converges uniformly on a compact subset of
2148:
5361:
The most general scenario in which equicontinuity can be defined is for
4542:
is an equicontinuous sequence of continuous functions on a dense subset
576:
depending on the context. On a compact space, these notions coincide.
6275:
6273:
6271:
6269:
6267:
5238:(0) = 1, and consider the equicontinuous sequence of functions {
316:
are continuous' means that for every Îľ > 0, every
6356:
6354:
6352:
5834:
encapsulate all the information necessary to define things such as
5804:, we would still be able to determine whether or not two points of
5880:) to a broader category of topological spaces. In particular, to
5606:
We now briefly describe the basic idea underlying uniformities.
5288:
converges pointwise to 0 but does not converge uniformly to 0.
4085:
implies that the weak-* closure of an equicontinuous subset of
3924:(also known as the BanachâSteinhaus theorem) states that a set
4680:. By denseness and compactness, we can find a finite subset
2459:
is locally convex then this list may be extended to include:
6315:
6313:
6311:
6309:
6307:
6294:
6292:
6290:
6288:
4419:
4367:
4301:
4203:
4171:
4097:
3896:
3855:
3815:
is bounded in the topology of bounded convergence (that is,
3778:
3737:
3654:
3566:
3407:
3167:
2893:
657:
endowed with a topology making its operations continuous).
6091:- an analogue of a continuous function in discrete spaces.
5381:. Appropriate definitions in these cases are as follows:
5222:
example.) In the above, the hypothesis of compactness of
4554: > 0 be given. By equicontinuity, for each
6111:
Pages displaying short descriptions of redirect targets
5903:
of continuous functions between two topological spaces
5389:
of functions continuous between two topological spaces
4221:
correspond to each other by polarity (with respect to
6054: â Mathematical analysis of discontinuous points
4966:
4849:
4739:
4584:
4408:
4388:
4356:
4332:
4290:
4270:
4227:
4192:
4165:
4145:
4121:
4091:
4056:
4036:
4007:
3950:
3930:
3890:
3841:
3821:
3801:
3772:
3723:
3703:
3679:
3648:
3620:
3593:
3560:
3540:
3513:
3489:
3462:
3435:
3401:
3381:
3352:
3332:
3306:
3282:
3258:
3238:
3216:
3194:
3155:
3124:
3068:
3048:
3028:
3006:
2986:
2960:
2937:
2917:
2887:
2862:
2838:
2826:
Characterization of equicontinuous linear functionals
2803:
2750:
2723:
2703:
2675:
2640:
2598:
2552:
2532:
2488:
2468:
2445:
2421:
2387:
2334:
2302:
2271:
2239:
2219:
2199:
2176:
2156:
2125:
2105:
2081:
2054:
2022:
1998:
1954:
1912:
1868:
1848:
1823:
1768:
1746:
1726:
1695:
1660:
1625:
1605:
1585:
1562:
1542:
1519:
1495:
1475:
1451:
1431:
1409:
1383:
1363:
1343:
1319:
1299:
1259:
1233:
1198:
1178:
1158:
1095:
1075:
1055:
1024:
968:
948:
928:
908:
888:
862:
833:
813:
480:
6618:
Topological Vector Spaces, Distributions and
Kernels
6123: â Uniform restraint of the change in functions
6073:
Pages displaying wikidata descriptions as a fallback
6065: â Mathematical function with no sudden changes
1983:
endowed with the topology of point-wise convergence.
853:
between two topological vector spaces is said to be
564:. This definition usually appears in the context of
5524:of continuous functions between two uniform spaces
4902:{\displaystyle \sup _{X}|f_{j}-f_{k}|<\epsilon }
4284:is barreled if and only if every bounded subset of
4257:{\displaystyle \left\langle X,X^{\#}\right\rangle }
637:
Equicontinuity of maps valued in topological groups
335:, there exists a δ > 0 such that
6109: â Function defined by multiple sub-functions
5188:
4901:
4808:{\displaystyle |f_{j}(z)-f_{k}(z)|<\epsilon /3}
4807:
4653:{\displaystyle |f_{j}(x)-f_{j}(z)|<\epsilon /3}
4652:
4426:
4394:
4374:
4338:
4308:
4276:
4256:
4213:
4186:that are convex, balanced, closed, and bounded in
4178:
4151:
4135:is any locally convex TVS, then the family of all
4127:
4104:
4062:
4042:
4022:
3993:
3936:
3903:
3876:
3827:
3807:
3785:
3759:{\displaystyle \sigma \left(X^{\prime },X\right)-}
3758:
3709:
3685:
3664:
3626:
3599:
3576:
3546:
3519:
3495:
3468:
3441:
3414:
3387:
3361:
3338:
3312:
3288:
3267:
3244:
3222:
3200:
3177:
3139:
3110:
3054:
3034:
3014:
2992:
2972:
2943:
2923:
2903:
2870:
2844:
2809:
2789:
2729:
2709:
2684:
2661:
2626:
2583:
2538:
2516:
2474:
2451:
2427:
2402:
2373:
2320:
2286:
2257:
2225:
2205:
2185:
2162:
2131:
2111:
2087:
2060:
2028:
2004:
1975:
1940:
1896:
1854:
1832:
1809:
1755:
1732:
1707:
1681:
1646:
1611:
1591:
1571:
1548:
1525:
1504:
1481:
1460:
1437:
1415:
1392:
1369:
1349:
1325:
1305:
1283:
1245:
1219:
1184:
1164:
1144:
1081:
1061:
1039:
1010:
954:
934:
914:
894:
874:
845:
819:
532:
6279:
6246:
3994:{\displaystyle \sup _{h\in H}\|h(x)\|<\infty }
71:continuous functions on a compact Hausdorff space
4851:
4506:In view of ArzelĂ âAscoli theorem, a sequence in
4350:TVS. Then every closed equicontinuous subset of
3952:
2751:
5793:-close. Note that if we were to "forget" that
5789:denote the set of all pairs of points that are
4078:Properties of equicontinuous linear functionals
2790:{\displaystyle \sup\{\|T\|:T\in H\}<\infty }
773:is equicontinuous at a point then every map in
6484:Narici, Lawrence; Beckenstein, Edward (2011).
803:Characterization of equicontinuous linear maps
6465:(revised and enlarged ed.), Boston, MA:
6360:
5665:Uniformities generalize the idea (taken from
5344:converges to a multiple of the discontinuous
4030:The result can be generalized to a case when
533:{\displaystyle d_{Y}(f(y),f(x))<\epsilon }
308:For comparison, the statement 'all functions
57:Equicontinuity appears in the formulation of
8:
3982:
3967:
3877:{\displaystyle b\left(X^{\prime },X\right)-}
2778:
2763:
2757:
2754:
2741:then this list may be extended to include:
2143:then this list may be extended to include:
1145:{\displaystyle H(U):=\bigcup _{h\in H}h(U).}
6103: â Function of ordinals in mathematics
5892:A weaker concept is that of even continuity
4320:
3611:then this list may be extended to include:
3531:then this list may be extended to include:
2040:then this list may be extended to include:
42:and they have equal variation over a given
6421:: CS1 maint: location missing publisher (
5623:where, among many other properties, every
5399:topologically equicontinuous at the points
1810:{\displaystyle \bigcap _{h\in H}h^{-1}(V)}
761:if it is equicontinuous at every point of
6184:
6168:
5479:if it is topologically equicontinuous at
5175:
5160:
5138:
5129:
5121:
5106:
5084:
5075:
5067:
5052:
5030:
5021:
5013:
4998:
4976:
4967:
4965:
4888:
4882:
4869:
4860:
4854:
4848:
4797:
4786:
4771:
4749:
4740:
4738:
4642:
4631:
4616:
4594:
4585:
4583:
4418:
4413:
4407:
4387:
4366:
4361:
4355:
4331:
4300:
4295:
4289:
4269:
4264:). It follows that a locally convex TVS
4243:
4226:
4202:
4197:
4191:
4170:
4164:
4144:
4120:
4096:
4090:
4055:
4035:
4006:
3955:
3949:
3929:
3895:
3889:
3854:
3840:
3820:
3800:
3777:
3771:
3736:
3722:
3702:
3678:
3653:
3647:
3619:
3592:
3565:
3559:
3539:
3512:
3488:
3461:
3434:
3406:
3400:
3380:
3351:
3331:
3305:
3281:
3257:
3237:
3215:
3193:
3166:
3154:
3123:
3067:
3047:
3027:
3008:
3007:
3005:
2985:
2959:
2936:
2916:
2892:
2886:
2864:
2863:
2861:
2837:
2802:
2749:
2722:
2702:
2674:
2639:
2603:
2597:
2557:
2551:
2531:
2493:
2487:
2467:
2444:
2420:
2386:
2333:
2301:
2270:
2238:
2218:
2198:
2175:
2155:
2124:
2104:
2080:
2053:
2021:
1997:
1953:
1917:
1911:
1873:
1867:
1847:
1822:
1789:
1773:
1767:
1745:
1725:
1694:
1659:
1624:
1604:
1584:
1561:
1541:
1518:
1494:
1474:
1450:
1430:
1408:
1382:
1362:
1342:
1318:
1298:
1258:
1232:
1197:
1177:
1157:
1115:
1094:
1074:
1054:
1023:
967:
947:
927:
907:
887:
861:
832:
812:
485:
479:
220:if it is equicontinuous at each point of
6343:
6331:
6048: â Form of continuity for functions
5613:is a non-empty collection of subsets of
4447:be a compact Hausdorff space, and equip
3916:Properties of equicontinuous linear maps
145:the respective metrics of these spaces.
6132:
5680:), meaning that their distance is <
3178:{\displaystyle H\subseteq X^{\prime },}
411:, δ may depend on ξ and
6414:
6319:
6298:
5696:is a metric space (so the diagonal of
5217:) and thus converges by completeness.
4439:Equicontinuity and uniform convergence
4214:{\displaystyle X_{\sigma }^{\prime },}
3300:of some neighborhood of the origin in
3111:{\displaystyle h(x+U)\subseteq h(x)+V}
1011:{\displaystyle h(x+U)\subseteq h(x)+V}
106:, then the limit is also holomorphic.
6620:. Mineola, N.Y.: Dover Publications.
6384:
6372:
6258:
6172:
6156:
4427:{\displaystyle X_{\sigma }^{\prime }}
4375:{\displaystyle X_{\sigma }^{\prime }}
4309:{\displaystyle X_{\sigma }^{\prime }}
1357:be a family of linear operators from
7:
6524:McGraw-Hill Science/Engineering/Math
5357:Equicontinuity in topological spaces
1445:is equicontinuous at every point of
1400:Then the following are equivalent:
113:Equicontinuity between metric spaces
3346:is a neighborhood of the origin in
3252:is equicontinuous at some point of
2193:there exists a continuous seminorm
1817:is a neighborhood of the origin in
1489:is equicontinuous at some point of
436:, δ may depend only on ξ.
216:) < δ. The family is
27:Relation among continuous functions
5588:is a member of the uniformity on
4244:
3988:
2784:
460:if for every Îľ > 0,
422:, δ may depend on ξ and
25:
6574:, New York: The Macmillan Company
6116:Symmetrically continuous function
6052:Classification of discontinuities
5684:. To clarify this, suppose that
4159:and the family of all subsets of
1533:is equicontinuous at the origin.
709:, there exists some neighborhood
593:A set of functions with a common
189:)) < Îľ for all
6069:Continuous function (set theory)
3554:is a strongly bounded subset of
3230:is equicontinuous at the origin.
2517:{\displaystyle L_{\sigma }(X;Y)}
2374:{\displaystyle q(h(x))\leq p(x)}
1941:{\displaystyle L_{\sigma }(X;Y)}
1897:{\displaystyle L_{\sigma }(X;Y)}
1536:that is, for every neighborhood
1284:{\displaystyle H(U)\subseteq V.}
456:is said to be equicontinuous at
95:is continuous. If, in addition,
61:, which states that a subset of
5470:topologically equicontinuous at
4487:, which states that subsets of
3185:the following are equivalent:
3022:there exists some neighborhood
1682:{\displaystyle h(U)\subseteq V}
1647:{\displaystyle H(U)\subseteq V}
1220:{\displaystyle h(U)\subseteq V}
922:there exists some neighborhood
391:, δ may depend on ξ,
6653:Theory of continuous functions
6001:if it is evenly continuous at
5981:if it is evenly continuous at
5812:-close by using only the sets
5596:Introduction to uniform spaces
5176:
5172:
5166:
5150:
5144:
5130:
5122:
5118:
5112:
5096:
5090:
5076:
5068:
5064:
5058:
5042:
5036:
5022:
5014:
5010:
5004:
4988:
4982:
4968:
4889:
4861:
4787:
4783:
4777:
4761:
4755:
4741:
4632:
4628:
4622:
4606:
4600:
4586:
4564:, there exists a neighborhood
4471:, hence a metric space. Then
3979:
3973:
3099:
3093:
3084:
3072:
2656:
2644:
2621:
2609:
2575:
2563:
2511:
2499:
2368:
2362:
2353:
2350:
2344:
2338:
2321:{\displaystyle q\circ h\leq p}
2258:{\displaystyle q\circ h\leq p}
1970:
1958:
1935:
1923:
1891:
1879:
1804:
1798:
1670:
1664:
1635:
1629:
1269:
1263:
1208:
1202:
1136:
1130:
1105:
1099:
999:
993:
984:
972:
837:
521:
518:
512:
503:
497:
491:
444:is a topological space, a set
1:
6401:. Amsterdam. pp. 53â72.
6280:Narici & Beckenstein 2011
6247:Narici & Beckenstein 2011
6089:Direction-preserving function
6078:Continuous stochastic process
3922:uniform boundedness principle
601:Uniform boundedness principle
6582:; Wolff, Manfred P. (1999).
5369:equicontinuity requires the
3665:{\displaystyle X^{\prime }.}
3577:{\displaystyle X^{\prime }.}
3015:{\displaystyle \mathbb {F} }
2904:{\displaystyle X^{\prime }.}
2871:{\displaystyle \mathbb {F} }
2817:is uniformly bounded in the
1579:there exists a neighborhood
358:)) < Îľ for all
261:)) < Îľ for all
6445:Encyclopedia of Mathematics
5503:if it is equicontinuous at
4179:{\displaystyle X^{\prime }}
4105:{\displaystyle X^{\prime }}
3904:{\displaystyle X^{\prime }}
3786:{\displaystyle X^{\prime }}
3415:{\displaystyle X^{\prime }}
2584:{\displaystyle L_{b}(X;Y).}
645:is a topological space and
624:The sequence of functions f
610:is equicontinuous on the
606:A family of iterates of an
133:a family of functions from
34:, a family of functions is
6674:
6558:(3rd ed.), New York:
6387:, p. 18 Theorem 1.23.
6375:, p. 394 Appendix A5.
6021:
5599:
5424:, there are neighborhoods
4526:(not assumed continuous).
2980:if for every neighborhood
2627:{\displaystyle L_{b}(X;Y)}
882:if for every neighborhood
789:Equicontinuous linear maps
697:if for every neighborhood
91:on either metric space or
18:Equicontinuous linear maps
6584:Topological Vector Spaces
6572:Topological vector spaces
6556:Real and Complex Analysis
6486:Topological Vector Spaces
6361:Schaefer & Wolff 1999
6261:, p. 44 Theorem 2.4.
6141:compactly generated space
6097: â Mathematical term
6024:Stochastic equicontinuity
6018:Stochastic equicontinuity
5886:topological vector spaces
5823:. In this way, the sets
5634:contains the diagonal of
5448:, if the intersection of
2953:equicontinuous at a point
2931:of linear functionals on
1335:topological vector spaces
855:equicontinuous at a point
659:Topological vector spaces
566:topological vector spaces
154:equicontinuous at a point
38:if all the functions are
5938:there are neighborhoods
5534:uniformly equicontinuous
4475:states that a subset of
3613:
3533:
2854:topological vector space
2743:
2461:
2145:
1069:is a family of maps and
797:topological vector space
420:pointwise equicontinuity
233:uniformly equicontinuous
218:pointwise equicontinuous
6185:Reed & Simon (1980)
6169:Reed & Simon (1980)
6139:More generally, on any
4720:converges pointwise on
4023:{\displaystyle x\in X.}
3140:{\displaystyle h\in H.}
2403:{\displaystyle x\in X.}
2287:{\displaystyle h\in H.}
2042:
1720:for every neighborhood
1040:{\displaystyle h\in H.}
5930:if given any open set
5797:existed then, for any
5669:) of points that are "
5312:; thus, continuous on
5190:
4903:
4809:
4654:
4428:
4396:
4376:
4340:
4310:
4278:
4258:
4215:
4180:
4153:
4129:
4106:
4064:
4050:is locally convex and
4044:
4024:
3995:
3938:
3905:
3878:
3829:
3809:
3787:
3760:
3711:
3687:
3666:
3628:
3601:
3578:
3548:
3521:
3497:
3470:
3443:
3416:
3389:
3363:
3340:
3314:
3290:
3269:
3246:
3224:
3202:
3179:
3141:
3112:
3056:
3036:
3016:
2994:
2974:
2973:{\displaystyle x\in X}
2945:
2925:
2905:
2872:
2846:
2811:
2791:
2731:
2711:
2686:
2663:
2662:{\displaystyle L(X;Y)}
2628:
2585:
2540:
2518:
2476:
2453:
2429:
2404:
2375:
2322:
2288:
2259:
2227:
2207:
2187:
2164:
2133:
2113:
2089:
2062:
2030:
2006:
1977:
1976:{\displaystyle L(X;Y)}
1942:
1898:
1856:
1834:
1811:
1757:
1734:
1709:
1708:{\displaystyle h\in H}
1683:
1648:
1613:
1593:
1573:
1550:
1527:
1506:
1483:
1462:
1439:
1417:
1394:
1371:
1351:
1327:
1307:
1285:
1247:
1246:{\displaystyle h\in H}
1221:
1186:
1166:
1146:
1083:
1063:
1041:
1012:
956:
936:
916:
896:
876:
875:{\displaystyle x\in X}
847:
846:{\displaystyle X\to Y}
821:
534:
434:uniform equicontinuity
6658:Mathematical analysis
6145:first-countable space
5654:). Every element of
5540:of the uniformity on
5536:if for every element
5191:
4904:
4810:
4655:
4473:ArzelĂ âAscoli theorem
4429:
4397:
4377:
4341:
4311:
4279:
4259:
4216:
4181:
4154:
4130:
4107:
4065:
4045:
4025:
3996:
3939:
3906:
3879:
3830:
3810:
3788:
3761:
3712:
3688:
3667:
3629:
3602:
3579:
3549:
3522:
3498:
3471:
3444:
3417:
3390:
3364:
3341:
3315:
3291:
3270:
3247:
3225:
3203:
3180:
3142:
3113:
3057:
3037:
3017:
2995:
2975:
2946:
2926:
2906:
2880:continuous dual space
2873:
2856:(TVS) over the field
2847:
2812:
2792:
2732:
2712:
2687:
2664:
2629:
2586:
2541:
2519:
2477:
2454:
2430:
2405:
2376:
2323:
2289:
2260:
2228:
2208:
2188:
2165:
2147:for every continuous
2134:
2114:
2090:
2063:
2031:
2007:
1978:
1943:
1899:
1857:
1835:
1812:
1758:
1735:
1710:
1684:
1649:
1614:
1594:
1574:
1551:
1528:
1507:
1484:
1463:
1440:
1418:
1395:
1372:
1352:
1328:
1308:
1286:
1248:
1222:
1187:
1167:
1147:
1084:
1064:
1042:
1013:
957:
937:
917:
897:
877:
848:
822:
535:
440:More generally, when
141:. We shall denote by
93:locally compact space
32:mathematical analysis
5976:evenly continuous at
5913:evenly continuous at
5850:. Indeed, the sets
5440:such that for every
5416:if for any open set
4964:
4847:
4737:
4582:
4406:
4386:
4354:
4330:
4288:
4268:
4225:
4190:
4163:
4143:
4119:
4089:
4054:
4034:
4005:
3948:
3928:
3888:
3839:
3819:
3799:
3770:
3721:
3701:
3677:
3646:
3618:
3591:
3558:
3538:
3511:
3487:
3481:convex balanced hull
3460:
3433:
3399:
3379:
3350:
3330:
3304:
3296:is contained in the
3280:
3256:
3236:
3214:
3192:
3153:
3122:
3066:
3046:
3026:
3004:
2984:
2958:
2935:
2915:
2885:
2860:
2836:
2801:
2748:
2721:
2701:
2673:
2638:
2596:
2550:
2530:
2486:
2466:
2443:
2419:
2385:
2332:
2300:
2269:
2237:
2217:
2197:
2174:
2154:
2123:
2103:
2079:
2073:convex balanced hull
2052:
2020:
1996:
1952:
1910:
1904:is equicontinuous.
1866:
1846:
1821:
1766:
1744:
1724:
1693:
1658:
1623:
1603:
1583:
1560:
1540:
1517:
1493:
1473:
1449:
1429:
1407:
1381:
1361:
1341:
1317:
1297:
1257:
1231:
1196:
1176:
1156:
1093:
1073:
1053:
1022:
966:
946:
926:
906:
886:
860:
831:
827:of maps of the form
811:
582:uniformly continuous
478:
381:) < δ.
305:) < δ.
6580:Schaefer, Helmut H.
6568:Schaefer, Helmut H.
6519:Functional Analysis
6463:Functional Analysis
6363:, pp. 123â128.
6322:, pp. 346â350.
6301:, pp. 335â345.
6282:, pp. 225â273.
6249:, pp. 133â136.
6063:Continuous function
6046:Absolute continuity
5840:uniform convergence
4485:HeineâBorel theorem
4423:
4402:is metrizable then
4371:
4324: —
4316:is equicontinuous.
4305:
4207:
785:is equicontinuous.
464:has a neighborhood
50:families, and thus
6159:, p. 44 §2.5.
6121:Uniform continuity
5882:topological groups
5836:uniform continuity
5363:topological spaces
5306:mean value theorem
5186:
4899:
4859:
4840:. It follows that
4805:
4650:
4531:
4424:
4409:
4392:
4372:
4357:
4336:
4322:
4306:
4291:
4274:
4254:
4211:
4193:
4176:
4149:
4125:
4102:
4060:
4040:
4020:
3991:
3966:
3934:
3901:
3874:
3825:
3805:
3783:
3756:
3707:
3683:
3662:
3636:relatively compact
3624:
3597:
3574:
3544:
3517:
3503:is equicontinuous.
3493:
3476:is equicontinuous.
3466:
3449:is equicontinuous.
3439:
3422:is equicontinuous.
3412:
3385:
3362:{\displaystyle X.}
3359:
3336:
3310:
3286:
3268:{\displaystyle X.}
3265:
3242:
3220:
3208:is equicontinuous.
3198:
3175:
3137:
3108:
3052:
3032:
3012:
2990:
2970:
2941:
2921:
2901:
2868:
2842:
2807:
2787:
2727:
2707:
2685:{\displaystyle X.}
2682:
2659:
2624:
2581:
2536:
2514:
2472:
2449:
2425:
2400:
2371:
2318:
2284:
2255:
2223:
2203:
2186:{\displaystyle Y,}
2183:
2160:
2129:
2109:
2095:is equicontinuous.
2085:
2068:is equicontinuous.
2058:
2026:
2012:is equicontinuous.
2002:
1973:
1938:
1894:
1852:
1833:{\displaystyle X.}
1830:
1807:
1784:
1756:{\displaystyle Y,}
1753:
1730:
1705:
1679:
1654:(or equivalently,
1644:
1609:
1589:
1572:{\displaystyle Y,}
1569:
1546:
1523:
1505:{\displaystyle X.}
1502:
1479:
1461:{\displaystyle X.}
1458:
1435:
1423:is equicontinuous;
1413:
1393:{\displaystyle Y.}
1390:
1367:
1347:
1323:
1303:
1281:
1243:
1217:
1182:
1162:
1152:With notation, if
1142:
1126:
1089:is a set then let
1079:
1059:
1037:
1008:
952:
932:
912:
892:
872:
843:
817:
595:Lipschitz constant
530:
448:of functions from
408:uniform continuity
6627:978-0-486-45352-1
6597:978-1-4612-7155-0
6533:978-0-07-054236-5
6476:978-0-12-585050-6
6235:978-0-387-69903-5
6210:978-0-387-95151-5
5999:evenly continuous
5375:uniform structure
5318:Cauchy's estimate
4850:
4730:> 0 such that
4529:
4395:{\displaystyle X}
4339:{\displaystyle X}
4277:{\displaystyle X}
4152:{\displaystyle X}
4128:{\displaystyle X}
4083:Alaoglu's theorem
4063:{\displaystyle X}
4043:{\displaystyle Y}
3951:
3937:{\displaystyle H}
3828:{\displaystyle H}
3808:{\displaystyle H}
3710:{\displaystyle H}
3686:{\displaystyle H}
3627:{\displaystyle H}
3600:{\displaystyle X}
3547:{\displaystyle H}
3520:{\displaystyle X}
3496:{\displaystyle H}
3469:{\displaystyle H}
3442:{\displaystyle H}
3388:{\displaystyle H}
3339:{\displaystyle H}
3313:{\displaystyle X}
3289:{\displaystyle H}
3245:{\displaystyle H}
3223:{\displaystyle H}
3201:{\displaystyle H}
3055:{\displaystyle X}
3042:of the origin in
3035:{\displaystyle U}
3000:of the origin in
2993:{\displaystyle V}
2944:{\displaystyle X}
2924:{\displaystyle H}
2845:{\displaystyle X}
2810:{\displaystyle H}
2730:{\displaystyle Y}
2710:{\displaystyle X}
2539:{\displaystyle H}
2475:{\displaystyle H}
2452:{\displaystyle Y}
2428:{\displaystyle X}
2226:{\displaystyle X}
2206:{\displaystyle p}
2163:{\displaystyle q}
2132:{\displaystyle Y}
2112:{\displaystyle X}
2088:{\displaystyle H}
2061:{\displaystyle H}
2029:{\displaystyle Y}
2005:{\displaystyle H}
1855:{\displaystyle H}
1769:
1740:of the origin in
1733:{\displaystyle V}
1612:{\displaystyle X}
1599:of the origin in
1592:{\displaystyle U}
1556:of the origin in
1549:{\displaystyle V}
1526:{\displaystyle H}
1482:{\displaystyle H}
1438:{\displaystyle H}
1416:{\displaystyle H}
1370:{\displaystyle X}
1350:{\displaystyle H}
1326:{\displaystyle Y}
1306:{\displaystyle X}
1185:{\displaystyle V}
1165:{\displaystyle U}
1111:
1082:{\displaystyle U}
1062:{\displaystyle H}
955:{\displaystyle X}
942:of the origin in
935:{\displaystyle U}
915:{\displaystyle Y}
902:of the origin in
895:{\displaystyle V}
820:{\displaystyle H}
686:equicontinuous at
651:topological group
608:analytic function
16:(Redirected from
6665:
6639:
6614:Trèves, François
6609:
6575:
6562:
6545:
6507:
6479:
6453:
6440:"Equicontinuity"
6427:
6426:
6420:
6412:
6394:
6388:
6382:
6376:
6370:
6364:
6358:
6347:
6346:, Corollary 4.3.
6341:
6335:
6329:
6323:
6317:
6302:
6296:
6283:
6277:
6262:
6256:
6250:
6244:
6238:
6219:
6213:
6194:
6188:
6182:
6176:
6166:
6160:
6154:
6148:
6137:
6112:
6074:
6030:random variables
5872:
5860:
5833:
5822:
5811:
5807:
5803:
5796:
5792:
5785:
5738:
5731:
5699:
5695:
5683:
5679:
5672:
5657:
5653:
5637:
5633:
5629:
5622:
5612:
5343:
5278:
5195:
5193:
5192:
5187:
5179:
5165:
5164:
5143:
5142:
5133:
5125:
5111:
5110:
5089:
5088:
5079:
5071:
5057:
5056:
5035:
5034:
5025:
5017:
5003:
5002:
4981:
4980:
4971:
4956:
4946:
4933:
4908:
4906:
4905:
4900:
4892:
4887:
4886:
4874:
4873:
4864:
4858:
4827:
4814:
4812:
4811:
4806:
4801:
4790:
4776:
4775:
4754:
4753:
4744:
4725:
4710:
4694:is the union of
4689:
4679:
4659:
4657:
4656:
4651:
4646:
4635:
4621:
4620:
4599:
4598:
4589:
4563:
4433:
4431:
4430:
4425:
4422:
4417:
4401:
4399:
4398:
4393:
4381:
4379:
4378:
4373:
4370:
4365:
4345:
4343:
4342:
4337:
4325:
4315:
4313:
4312:
4307:
4304:
4299:
4283:
4281:
4280:
4275:
4263:
4261:
4260:
4255:
4253:
4249:
4248:
4247:
4220:
4218:
4217:
4212:
4206:
4201:
4185:
4183:
4182:
4177:
4175:
4174:
4158:
4156:
4155:
4150:
4134:
4132:
4131:
4126:
4111:
4109:
4108:
4103:
4101:
4100:
4069:
4067:
4066:
4061:
4049:
4047:
4046:
4041:
4029:
4027:
4026:
4021:
4000:
3998:
3997:
3992:
3965:
3943:
3941:
3940:
3935:
3910:
3908:
3907:
3902:
3900:
3899:
3883:
3881:
3880:
3875:
3870:
3866:
3859:
3858:
3834:
3832:
3831:
3826:
3814:
3812:
3811:
3806:
3792:
3790:
3789:
3784:
3782:
3781:
3765:
3763:
3762:
3757:
3752:
3748:
3741:
3740:
3716:
3714:
3713:
3708:
3692:
3690:
3689:
3684:
3671:
3669:
3668:
3663:
3658:
3657:
3633:
3631:
3630:
3625:
3606:
3604:
3603:
3598:
3583:
3581:
3580:
3575:
3570:
3569:
3553:
3551:
3550:
3545:
3526:
3524:
3523:
3518:
3502:
3500:
3499:
3494:
3475:
3473:
3472:
3467:
3448:
3446:
3445:
3440:
3421:
3419:
3418:
3413:
3411:
3410:
3394:
3392:
3391:
3386:
3368:
3366:
3365:
3360:
3345:
3343:
3342:
3337:
3319:
3317:
3316:
3311:
3295:
3293:
3292:
3287:
3274:
3272:
3271:
3266:
3251:
3249:
3248:
3243:
3229:
3227:
3226:
3221:
3207:
3205:
3204:
3199:
3184:
3182:
3181:
3176:
3171:
3170:
3146:
3144:
3143:
3138:
3117:
3115:
3114:
3109:
3061:
3059:
3058:
3053:
3041:
3039:
3038:
3033:
3021:
3019:
3018:
3013:
3011:
2999:
2997:
2996:
2991:
2979:
2977:
2976:
2971:
2950:
2948:
2947:
2942:
2930:
2928:
2927:
2922:
2910:
2908:
2907:
2902:
2897:
2896:
2877:
2875:
2874:
2869:
2867:
2851:
2849:
2848:
2843:
2816:
2814:
2813:
2808:
2796:
2794:
2793:
2788:
2736:
2734:
2733:
2728:
2716:
2714:
2713:
2708:
2691:
2689:
2688:
2683:
2668:
2666:
2665:
2660:
2633:
2631:
2630:
2625:
2608:
2607:
2590:
2588:
2587:
2582:
2562:
2561:
2545:
2543:
2542:
2537:
2523:
2521:
2520:
2515:
2498:
2497:
2481:
2479:
2478:
2473:
2458:
2456:
2455:
2450:
2434:
2432:
2431:
2426:
2409:
2407:
2406:
2401:
2380:
2378:
2377:
2372:
2327:
2325:
2324:
2319:
2293:
2291:
2290:
2285:
2264:
2262:
2261:
2256:
2232:
2230:
2229:
2224:
2212:
2210:
2209:
2204:
2192:
2190:
2189:
2184:
2169:
2167:
2166:
2161:
2138:
2136:
2135:
2130:
2118:
2116:
2115:
2110:
2094:
2092:
2091:
2086:
2067:
2065:
2064:
2059:
2035:
2033:
2032:
2027:
2011:
2009:
2008:
2003:
1982:
1980:
1979:
1974:
1947:
1945:
1944:
1939:
1922:
1921:
1903:
1901:
1900:
1895:
1878:
1877:
1861:
1859:
1858:
1853:
1839:
1837:
1836:
1831:
1816:
1814:
1813:
1808:
1797:
1796:
1783:
1762:
1760:
1759:
1754:
1739:
1737:
1736:
1731:
1714:
1712:
1711:
1706:
1688:
1686:
1685:
1680:
1653:
1651:
1650:
1645:
1618:
1616:
1615:
1610:
1598:
1596:
1595:
1590:
1578:
1576:
1575:
1570:
1555:
1553:
1552:
1547:
1532:
1530:
1529:
1524:
1511:
1509:
1508:
1503:
1488:
1486:
1485:
1480:
1467:
1465:
1464:
1459:
1444:
1442:
1441:
1436:
1422:
1420:
1419:
1414:
1399:
1397:
1396:
1391:
1376:
1374:
1373:
1368:
1356:
1354:
1353:
1348:
1332:
1330:
1329:
1324:
1312:
1310:
1309:
1304:
1290:
1288:
1287:
1282:
1252:
1250:
1249:
1244:
1226:
1224:
1223:
1218:
1191:
1189:
1188:
1183:
1171:
1169:
1168:
1163:
1151:
1149:
1148:
1143:
1125:
1088:
1086:
1085:
1080:
1068:
1066:
1065:
1060:
1046:
1044:
1043:
1038:
1017:
1015:
1014:
1009:
961:
959:
958:
953:
941:
939:
938:
933:
921:
919:
918:
913:
901:
899:
898:
893:
881:
879:
878:
873:
852:
850:
849:
844:
826:
824:
823:
818:
784:
780:
776:
772:
764:
756:
752:
742:
720:
716:
712:
708:
704:
700:
696:
683:
679:
675:
648:
644:
555:
539:
537:
536:
531:
490:
489:
227:The family
148:The family
69:), the space of
59:Ascoli's theorem
21:
6673:
6672:
6668:
6667:
6666:
6664:
6663:
6662:
6643:
6642:
6628:
6612:
6598:
6578:
6566:
6550:
6534:
6512:
6496:
6483:
6477:
6457:Reed, Michael;
6456:
6438:
6435:
6430:
6413:
6409:
6396:
6395:
6391:
6383:
6379:
6371:
6367:
6359:
6350:
6342:
6338:
6330:
6326:
6318:
6305:
6297:
6286:
6278:
6265:
6257:
6253:
6245:
6241:
6220:
6216:
6195:
6191:
6183:
6179:
6167:
6163:
6155:
6151:
6138:
6134:
6130:
6110:
6101:Normal function
6095:Microcontinuity
6084:Dini continuity
6072:
6058:Coarse function
6042:
6026:
6020:
5862:
5859:
5851:
5832:
5824:
5821:
5813:
5809:
5805:
5798:
5794:
5790:
5751:
5743:
5733:
5701:
5697:
5685:
5681:
5674:
5670:
5655:
5639:
5635:
5631:
5624:
5614:
5610:
5609:The uniformity
5604:
5507:for all points
5359:
5354:
5352:Generalizations
5334:
5328:
5287:
5265:
5259:
5246:
5219:
5208:
5156:
5134:
5102:
5080:
5048:
5026:
4994:
4972:
4962:
4961:
4957:and so we get:
4948:
4944:
4935:
4925:
4878:
4865:
4845:
4844:
4819:
4767:
4745:
4735:
4734:
4726:, there exists
4721:
4719:
4702:
4699:
4681:
4677:
4668:
4612:
4590:
4580:
4579:
4570:
4555:
4541:
4441:
4436:
4404:
4403:
4384:
4383:
4352:
4351:
4328:
4327:
4323:
4286:
4285:
4266:
4265:
4239:
4232:
4228:
4223:
4222:
4188:
4187:
4166:
4161:
4160:
4141:
4140:
4117:
4116:
4092:
4087:
4086:
4080:
4052:
4051:
4032:
4031:
4003:
4002:
3946:
3945:
3926:
3925:
3918:
3891:
3886:
3885:
3850:
3849:
3845:
3837:
3836:
3817:
3816:
3797:
3796:
3773:
3768:
3767:
3732:
3731:
3727:
3719:
3718:
3699:
3698:
3675:
3674:
3649:
3644:
3643:
3616:
3615:
3589:
3588:
3561:
3556:
3555:
3536:
3535:
3509:
3508:
3485:
3484:
3458:
3457:
3431:
3430:
3402:
3397:
3396:
3377:
3376:
3348:
3347:
3328:
3327:
3302:
3301:
3278:
3277:
3254:
3253:
3234:
3233:
3212:
3211:
3190:
3189:
3162:
3151:
3150:
3149:For any subset
3120:
3119:
3064:
3063:
3044:
3043:
3024:
3023:
3002:
3001:
2982:
2981:
2956:
2955:
2933:
2932:
2913:
2912:
2888:
2883:
2882:
2858:
2857:
2834:
2833:
2828:
2799:
2798:
2746:
2745:
2719:
2718:
2699:
2698:
2671:
2670:
2636:
2635:
2599:
2594:
2593:
2553:
2548:
2547:
2528:
2527:
2489:
2484:
2483:
2464:
2463:
2441:
2440:
2417:
2416:
2383:
2382:
2330:
2329:
2298:
2297:
2267:
2266:
2235:
2234:
2215:
2214:
2195:
2194:
2172:
2171:
2152:
2151:
2121:
2120:
2101:
2100:
2077:
2076:
2050:
2049:
2018:
2017:
1994:
1993:
1950:
1949:
1913:
1908:
1907:
1869:
1864:
1863:
1844:
1843:
1842:the closure of
1819:
1818:
1785:
1764:
1763:
1742:
1741:
1722:
1721:
1691:
1690:
1656:
1655:
1621:
1620:
1601:
1600:
1581:
1580:
1558:
1557:
1538:
1537:
1515:
1514:
1491:
1490:
1471:
1470:
1447:
1446:
1427:
1426:
1405:
1404:
1379:
1378:
1359:
1358:
1339:
1338:
1315:
1314:
1295:
1294:
1255:
1254:
1253:if and only if
1229:
1228:
1194:
1193:
1174:
1173:
1154:
1153:
1091:
1090:
1071:
1070:
1051:
1050:
1020:
1019:
964:
963:
944:
943:
924:
923:
904:
903:
884:
883:
858:
857:
829:
828:
809:
808:
805:
791:
782:
778:
774:
770:
762:
754:
744:
722:
718:
714:
710:
706:
702:
698:
688:
681:
677:
673:
649:is an additive
646:
642:
639:
631:
627:
621:
619:Counterexamples
590:
553:
544:
481:
476:
475:
469:
428:
401:
376:
349:
330:
304:
297:
282:
275:
260:
249:
211:
180:
161:
115:
100:
89:
28:
23:
22:
15:
12:
11:
5:
6671:
6669:
6661:
6660:
6655:
6645:
6644:
6641:
6640:
6626:
6610:
6596:
6576:
6564:
6547:
6546:
6532:
6509:
6508:
6495:978-1584888666
6494:
6481:
6475:
6467:Academic Press
6454:
6434:
6431:
6429:
6428:
6407:
6389:
6377:
6365:
6348:
6336:
6334:, Theorem 4.2.
6324:
6303:
6284:
6263:
6251:
6239:
6214:
6189:
6177:
6161:
6149:
6131:
6129:
6126:
6125:
6124:
6118:
6113:
6104:
6098:
6092:
6086:
6081:
6075:
6066:
6060:
6055:
6049:
6041:
6038:
6022:Main article:
6019:
6016:
6015:
6014:
5911:is said to be
5896:
5895:
5893:
5855:
5828:
5817:
5787:
5786:
5747:
5600:Main article:
5598:
5597:
5593:
5592:
5586:
5585:
5584:
5517:
5516:
5501:equicontinuous
5468:is said to be
5456:is nonempty,
5358:
5355:
5353:
5350:
5324:
5283:
5255:
5242:
5204:
5198:
5197:
5185:
5182:
5178:
5174:
5171:
5168:
5163:
5159:
5155:
5152:
5149:
5146:
5141:
5137:
5132:
5128:
5124:
5120:
5117:
5114:
5109:
5105:
5101:
5098:
5095:
5092:
5087:
5083:
5078:
5074:
5070:
5066:
5063:
5060:
5055:
5051:
5047:
5044:
5041:
5038:
5033:
5029:
5024:
5020:
5016:
5012:
5009:
5006:
5001:
4997:
4993:
4990:
4987:
4984:
4979:
4975:
4970:
4942:
4924:. In fact, if
4910:
4909:
4898:
4895:
4891:
4885:
4881:
4877:
4872:
4868:
4863:
4857:
4853:
4816:
4815:
4804:
4800:
4796:
4793:
4789:
4785:
4782:
4779:
4774:
4770:
4766:
4763:
4760:
4757:
4752:
4748:
4743:
4715:
4697:
4675:
4661:
4660:
4649:
4645:
4641:
4638:
4634:
4630:
4627:
4624:
4619:
4615:
4611:
4608:
4605:
4602:
4597:
4593:
4588:
4568:
4537:
4528:
4459:, thus making
4440:
4437:
4434:is separable.
4421:
4416:
4412:
4391:
4369:
4364:
4360:
4335:
4318:
4303:
4298:
4294:
4273:
4252:
4246:
4242:
4238:
4235:
4231:
4210:
4205:
4200:
4196:
4173:
4169:
4148:
4124:
4099:
4095:
4079:
4076:
4072:barreled space
4059:
4039:
4019:
4016:
4013:
4010:
3990:
3987:
3984:
3981:
3978:
3975:
3972:
3969:
3964:
3961:
3958:
3954:
3933:
3917:
3914:
3913:
3912:
3898:
3894:
3873:
3869:
3865:
3862:
3857:
3853:
3848:
3844:
3824:
3804:
3794:
3780:
3776:
3755:
3751:
3747:
3744:
3739:
3735:
3730:
3726:
3706:
3682:
3672:
3661:
3656:
3652:
3640:weak* topology
3623:
3609:barreled space
3596:
3585:
3584:
3573:
3568:
3564:
3543:
3516:
3505:
3504:
3492:
3477:
3465:
3450:
3438:
3423:
3409:
3405:
3384:
3369:
3358:
3355:
3335:
3320:
3309:
3285:
3275:
3264:
3261:
3241:
3231:
3219:
3209:
3197:
3174:
3169:
3165:
3161:
3158:
3136:
3133:
3130:
3127:
3107:
3104:
3101:
3098:
3095:
3092:
3089:
3086:
3083:
3080:
3077:
3074:
3071:
3051:
3031:
3010:
2989:
2969:
2966:
2963:
2954:
2951:is said to be
2940:
2920:
2900:
2895:
2891:
2866:
2841:
2827:
2824:
2823:
2822:
2806:
2786:
2783:
2780:
2777:
2774:
2771:
2768:
2765:
2762:
2759:
2756:
2753:
2726:
2706:
2695:
2694:
2693:
2692:
2681:
2678:
2658:
2655:
2652:
2649:
2646:
2643:
2623:
2620:
2617:
2614:
2611:
2606:
2602:
2580:
2577:
2574:
2571:
2568:
2565:
2560:
2556:
2546:is bounded in
2535:
2525:
2513:
2510:
2507:
2504:
2501:
2496:
2492:
2482:is bounded in
2471:
2448:
2424:
2413:
2412:
2411:
2410:
2399:
2396:
2393:
2390:
2370:
2367:
2364:
2361:
2358:
2355:
2352:
2349:
2346:
2343:
2340:
2337:
2317:
2314:
2311:
2308:
2305:
2283:
2280:
2277:
2274:
2254:
2251:
2248:
2245:
2242:
2222:
2202:
2182:
2179:
2159:
2141:locally convex
2128:
2108:
2097:
2096:
2084:
2069:
2057:
2038:locally convex
2025:
2014:
2013:
2001:
1986:
1985:
1984:
1972:
1969:
1966:
1963:
1960:
1957:
1937:
1934:
1931:
1928:
1925:
1920:
1916:
1893:
1890:
1887:
1884:
1881:
1876:
1872:
1851:
1840:
1829:
1826:
1806:
1803:
1800:
1795:
1792:
1788:
1782:
1779:
1776:
1772:
1752:
1749:
1729:
1718:
1717:
1716:
1704:
1701:
1698:
1678:
1675:
1672:
1669:
1666:
1663:
1643:
1640:
1637:
1634:
1631:
1628:
1608:
1588:
1568:
1565:
1545:
1522:
1512:
1501:
1498:
1478:
1468:
1457:
1454:
1434:
1424:
1412:
1389:
1386:
1366:
1346:
1322:
1302:
1280:
1277:
1274:
1271:
1268:
1265:
1262:
1242:
1239:
1236:
1216:
1213:
1210:
1207:
1204:
1201:
1192:are sets then
1181:
1161:
1141:
1138:
1135:
1132:
1129:
1124:
1121:
1118:
1114:
1110:
1107:
1104:
1101:
1098:
1078:
1058:
1036:
1033:
1030:
1027:
1007:
1004:
1001:
998:
995:
992:
989:
986:
983:
980:
977:
974:
971:
951:
931:
911:
891:
871:
868:
865:
856:
842:
839:
836:
816:
804:
801:
795:Because every
790:
787:
767:
766:
759:equicontinuous
753:. We say that
684:is said to be
638:
635:
634:
633:
629:
625:
620:
617:
616:
615:
604:
598:
589:
586:
551:
541:
540:
529:
526:
523:
520:
517:
514:
511:
508:
505:
502:
499:
496:
493:
488:
484:
467:
438:
437:
430:
426:
416:
403:
399:
374:
347:
328:
302:
295:
280:
273:
258:
247:
209:
178:
159:
114:
111:
98:
87:
54:of functions.
36:equicontinuous
26:
24:
14:
13:
10:
9:
6:
4:
3:
2:
6670:
6659:
6656:
6654:
6651:
6650:
6648:
6637:
6633:
6629:
6623:
6619:
6615:
6611:
6607:
6603:
6599:
6593:
6589:
6585:
6581:
6577:
6573:
6569:
6565:
6561:
6557:
6553:
6552:Rudin, Walter
6549:
6548:
6543:
6539:
6535:
6529:
6525:
6521:
6520:
6515:
6514:Rudin, Walter
6511:
6510:
6505:
6501:
6497:
6491:
6487:
6482:
6478:
6472:
6468:
6464:
6460:
6455:
6451:
6447:
6446:
6441:
6437:
6436:
6432:
6424:
6418:
6410:
6408:90-5170-227-2
6404:
6400:
6393:
6390:
6386:
6381:
6378:
6374:
6369:
6366:
6362:
6357:
6355:
6353:
6349:
6345:
6344:Schaefer 1966
6340:
6337:
6333:
6332:Schaefer 1966
6328:
6325:
6321:
6316:
6314:
6312:
6310:
6308:
6304:
6300:
6295:
6293:
6291:
6289:
6285:
6281:
6276:
6274:
6272:
6270:
6268:
6264:
6260:
6255:
6252:
6248:
6243:
6240:
6236:
6232:
6228:
6227:0-387-69903-1
6224:
6218:
6215:
6211:
6207:
6203:
6202:0-387-95151-2
6199:
6193:
6190:
6186:
6181:
6178:
6174:
6170:
6165:
6162:
6158:
6153:
6150:
6146:
6142:
6136:
6133:
6127:
6122:
6119:
6117:
6114:
6108:
6105:
6102:
6099:
6096:
6093:
6090:
6087:
6085:
6082:
6079:
6076:
6070:
6067:
6064:
6061:
6059:
6056:
6053:
6050:
6047:
6044:
6043:
6039:
6037:
6035:
6031:
6025:
6017:
6012:
6008:
6004:
6000:
5996:
5992:
5988:
5984:
5980:
5977:
5973:
5969:
5965:
5961:
5957:
5953:
5949:
5945:
5941:
5937:
5933:
5929:
5925:
5921:
5917:
5914:
5910:
5906:
5902:
5898:
5897:
5894:
5891:
5890:
5889:
5887:
5883:
5879:
5874:
5870:
5866:
5858:
5854:
5849:
5845:
5841:
5837:
5831:
5827:
5820:
5816:
5801:
5783:
5779:
5775:
5771:
5767:
5763:
5759:
5755:
5750:
5746:
5742:
5741:
5740:
5736:
5729:
5725:
5721:
5717:
5713:
5709:
5705:
5693:
5689:
5677:
5673:-close" (for
5668:
5667:metric spaces
5663:
5661:
5658:is called an
5651:
5647:
5643:
5627:
5621:
5617:
5607:
5603:
5602:Uniform space
5595:
5594:
5591:
5587:
5582:
5578:
5574:
5570:
5566:
5562:
5558:
5554:
5550:
5546:
5545:
5543:
5539:
5535:
5531:
5527:
5523:
5519:
5518:
5514:
5510:
5506:
5502:
5498:
5494:
5490:
5486:
5482:
5478:
5474:
5471:
5467:
5463:
5459:
5455:
5451:
5447:
5443:
5439:
5435:
5431:
5427:
5423:
5419:
5415:
5411:
5407:
5403:
5400:
5396:
5392:
5388:
5384:
5383:
5382:
5380:
5379:uniform space
5376:
5372:
5368:
5364:
5356:
5351:
5349:
5347:
5346:sign function
5342:
5338:
5332:
5327:
5323:
5319:
5315:
5311:
5307:
5302:
5298:
5294:
5289:
5286:
5282:
5276:
5272:
5268:
5263:
5258:
5254:
5250:
5245:
5241:
5237:
5233:
5229:
5225:
5218:
5216:
5212:
5209:is Cauchy in
5207:
5203:
5183:
5180:
5169:
5161:
5157:
5153:
5147:
5139:
5135:
5126:
5115:
5107:
5103:
5099:
5093:
5085:
5081:
5072:
5061:
5053:
5049:
5045:
5039:
5031:
5027:
5018:
5007:
4999:
4995:
4991:
4985:
4977:
4973:
4960:
4959:
4958:
4955:
4951:
4945:
4938:
4932:
4928:
4923:
4919:
4915:
4896:
4893:
4883:
4879:
4875:
4870:
4866:
4855:
4843:
4842:
4841:
4839:
4835:
4831:
4826:
4822:
4802:
4798:
4794:
4791:
4780:
4772:
4768:
4764:
4758:
4750:
4746:
4733:
4732:
4731:
4729:
4724:
4718:
4714:
4709:
4705:
4700:
4693:
4688:
4684:
4678:
4671:
4666:
4647:
4643:
4639:
4636:
4625:
4617:
4613:
4609:
4603:
4595:
4591:
4578:
4577:
4576:
4574:
4567:
4562:
4558:
4553:
4549:
4545:
4540:
4536:
4527:
4525:
4521:
4517:
4513:
4509:
4504:
4502:
4498:
4494:
4490:
4486:
4482:
4478:
4474:
4470:
4466:
4462:
4458:
4454:
4450:
4446:
4438:
4435:
4414:
4410:
4389:
4362:
4358:
4349:
4333:
4326:Suppose that
4317:
4296:
4292:
4271:
4250:
4240:
4236:
4233:
4229:
4208:
4198:
4194:
4167:
4146:
4138:
4122:
4113:
4093:
4084:
4077:
4075:
4073:
4057:
4037:
4017:
4014:
4011:
4008:
3985:
3976:
3970:
3962:
3959:
3956:
3931:
3923:
3915:
3892:
3871:
3867:
3863:
3860:
3851:
3846:
3842:
3822:
3802:
3795:
3774:
3753:
3749:
3745:
3742:
3733:
3728:
3724:
3704:
3696:
3695:weak* bounded
3680:
3673:
3659:
3650:
3641:
3637:
3621:
3614:
3612:
3610:
3594:
3571:
3562:
3541:
3534:
3532:
3530:
3514:
3490:
3482:
3478:
3463:
3455:
3451:
3436:
3428:
3427:balanced hull
3424:
3403:
3382:
3374:
3373:weak* closure
3370:
3356:
3353:
3333:
3325:
3321:
3307:
3299:
3283:
3276:
3262:
3259:
3239:
3232:
3217:
3210:
3195:
3188:
3187:
3186:
3172:
3163:
3159:
3156:
3147:
3134:
3131:
3128:
3125:
3105:
3102:
3096:
3090:
3087:
3081:
3078:
3075:
3069:
3049:
3029:
2987:
2967:
2964:
2961:
2952:
2938:
2918:
2898:
2889:
2881:
2855:
2839:
2830:
2825:
2820:
2819:operator norm
2804:
2781:
2775:
2772:
2769:
2766:
2760:
2744:
2742:
2740:
2739:Banach spaces
2724:
2704:
2679:
2676:
2653:
2650:
2647:
2641:
2618:
2615:
2612:
2604:
2600:
2592:
2591:
2578:
2572:
2569:
2566:
2558:
2554:
2533:
2526:
2508:
2505:
2502:
2494:
2490:
2469:
2462:
2460:
2446:
2438:
2422:
2397:
2394:
2391:
2388:
2365:
2359:
2356:
2347:
2341:
2335:
2315:
2312:
2309:
2306:
2303:
2295:
2294:
2281:
2278:
2275:
2272:
2252:
2249:
2246:
2243:
2240:
2220:
2200:
2180:
2177:
2157:
2150:
2146:
2144:
2142:
2126:
2106:
2082:
2074:
2070:
2055:
2047:
2043:
2041:
2039:
2023:
1999:
1991:
1990:balanced hull
1987:
1967:
1964:
1961:
1955:
1932:
1929:
1926:
1918:
1914:
1906:
1905:
1888:
1885:
1882:
1874:
1870:
1849:
1841:
1827:
1824:
1801:
1793:
1790:
1786:
1780:
1777:
1774:
1770:
1750:
1747:
1727:
1719:
1702:
1699:
1696:
1676:
1673:
1667:
1661:
1641:
1638:
1632:
1626:
1606:
1586:
1566:
1563:
1543:
1535:
1534:
1520:
1513:
1499:
1496:
1476:
1469:
1455:
1452:
1432:
1425:
1410:
1403:
1402:
1401:
1387:
1384:
1364:
1344:
1336:
1320:
1300:
1291:
1278:
1275:
1272:
1266:
1260:
1240:
1237:
1234:
1214:
1211:
1205:
1199:
1179:
1159:
1139:
1133:
1127:
1122:
1119:
1116:
1112:
1108:
1102:
1096:
1076:
1056:
1047:
1034:
1031:
1028:
1025:
1005:
1002:
996:
990:
987:
981:
978:
975:
969:
949:
929:
909:
889:
869:
866:
863:
854:
840:
834:
814:
802:
800:
798:
793:
788:
786:
769:Note that if
760:
751:
747:
741:
737:
733:
729:
725:
695:
691:
687:
676:of maps from
671:
668:
667:
666:
664:
660:
656:
652:
641:Suppose that
636:
623:
622:
618:
613:
609:
605:
602:
599:
596:
592:
591:
587:
585:
583:
577:
574:
569:
567:
563:
560: â
559:
554:
547:
527:
524:
515:
509:
506:
500:
494:
486:
482:
474:
473:
472:
470:
463:
459:
455:
451:
447:
443:
435:
431:
425:
421:
417:
414:
410:
409:
404:
398:
394:
390:
389:
384:
383:
382:
380:
373:
369:
365:
362: â
361:
357:
353:
346:
342:
338:
334:
331: â
327:
323:
320: â
319:
315:
311:
306:
301:
294:
290:
286:
283: â
279:
272:
268:
265: â
264:
257:
253:
246:
242:
238:
234:
230:
225:
223:
219:
215:
208:
204:
200:
196:
193: â
192:
188:
184:
177:
173:
169:
165:
162: â
158:
155:
151:
146:
144:
140:
136:
132:
128:
127:metric spaces
124:
120:
112:
110:
107:
105:
101:
94:
90:
83:
79:
75:
72:
68:
64:
60:
55:
53:
49:
45:
44:neighbourhood
41:
37:
33:
19:
6617:
6583:
6571:
6555:
6518:
6485:
6462:
6459:Simon, Barry
6443:
6398:
6392:
6380:
6368:
6339:
6327:
6254:
6242:
6217:
6192:
6180:
6173:Rudin (1987)
6164:
6152:
6135:
6032:, and their
6027:
6010:
6006:
6002:
5998:
5994:
5990:
5986:
5982:
5978:
5975:
5971:
5967:
5963:
5959:
5955:
5951:
5947:
5943:
5939:
5935:
5931:
5927:
5923:
5919:
5915:
5912:
5908:
5904:
5900:
5878:completeness
5875:
5868:
5864:
5856:
5852:
5843:
5829:
5825:
5818:
5814:
5799:
5788:
5781:
5777:
5773:
5769:
5765:
5761:
5757:
5753:
5748:
5744:
5734:
5727:
5723:
5719:
5715:
5711:
5707:
5703:
5691:
5687:
5675:
5664:
5659:
5649:
5645:
5641:
5625:
5619:
5615:
5608:
5605:
5589:
5580:
5576:
5572:
5568:
5564:
5560:
5556:
5552:
5548:
5541:
5537:
5533:
5529:
5525:
5521:
5512:
5508:
5504:
5500:
5496:
5492:
5488:
5484:
5480:
5476:
5472:
5469:
5465:
5461:
5457:
5453:
5449:
5445:
5441:
5437:
5433:
5429:
5425:
5421:
5417:
5413:
5409:
5405:
5401:
5398:
5394:
5390:
5386:
5366:
5360:
5340:
5336:
5330:
5325:
5321:
5313:
5309:
5300:
5296:
5292:
5290:
5284:
5280:
5274:
5270:
5266:
5261:
5256:
5252:
5248:
5243:
5239:
5235:
5231:
5227:
5223:
5220:
5214:
5210:
5205:
5201:
5199:
4953:
4949:
4940:
4936:
4930:
4926:
4921:
4917:
4913:
4911:
4837:
4833:
4829:
4824:
4820:
4817:
4727:
4722:
4716:
4712:
4707:
4703:
4695:
4691:
4686:
4682:
4673:
4669:
4664:
4662:
4572:
4565:
4560:
4556:
4551:
4547:
4543:
4538:
4534:
4532:
4523:
4519:
4515:
4511:
4507:
4505:
4500:
4496:
4492:
4488:
4480:
4476:
4469:Banach space
4464:
4460:
4457:uniform norm
4452:
4448:
4444:
4442:
4319:
4114:
4081:
3919:
3586:
3506:
3148:
2831:
2829:
2696:
2414:
2098:
2015:
1292:
1048:
806:
794:
792:
768:
758:
749:
745:
739:
735:
731:
727:
723:
693:
689:
685:
669:
640:
578:
572:
570:
561:
557:
549:
545:
542:
465:
461:
457:
453:
449:
445:
441:
439:
433:
423:
419:
412:
406:
396:
392:
386:
378:
371:
367:
363:
359:
355:
351:
344:
340:
336:
332:
325:
324:, and every
321:
317:
313:
309:
307:
299:
292:
288:
284:
277:
270:
266:
262:
255:
251:
244:
240:
236:
232:
228:
226:
221:
217:
213:
206:
202:
198:
194:
190:
186:
182:
175:
171:
167:
163:
156:
153:
149:
147:
142:
138:
134:
130:
122:
118:
116:
108:
96:
85:
81:
77:
73:
66:
62:
56:
51:
35:
29:
6560:McGraw-Hill
6320:Trèves 2006
6299:Trèves 2006
6034:convergence
5934:containing
5700:is the set
5544:, the set
5495:. Finally,
5377:, giving a
5251:defined by
4455:) with the
3884:bounded in
3766:bounded in
3454:convex hull
2328:means that
2046:convex hull
1337:(TVSs) and
672:: A family
104:holomorphic
6647:Categories
6433:References
6385:Rudin 1991
6373:Rudin 1991
6259:Rudin 1991
6157:Rudin 1991
6143:; e.g., a
6005:for every
5989:for every
5954:such that
5848:uniformity
5760:) ∈
5732:) For any
5710:) ∈
5648:) ∈
5628:∈ đą
5555:: for all
4690:such that
4575:such that
3697:(that is,
3324:(pre)polar
3062:such that
2797:(that is,
2233:such that
1689:for every
1619:such that
962:such that
743:for every
721:such that
670:Definition
663:uniformity
471:such that
388:continuity
366:such that
287:such that
201:such that
40:continuous
6636:853623322
6616:(2006) .
6606:840278135
6504:144216834
6450:EMS Press
6417:cite book
6237:; page 22
6212:; page 49
6171:, p. 29;
6107:Piecewise
5962:whenever
5660:entourage
5487:for each
5333:) =
5279:. Then,
5264:) =
5184:ϵ
5154:−
5100:−
5046:−
5019:≤
4992:−
4947:for some
4897:ϵ
4876:−
4818:whenever
4795:ϵ
4765:−
4640:ϵ
4610:−
4420:′
4415:σ
4368:′
4363:σ
4348:separable
4302:′
4297:σ
4245:#
4204:′
4199:σ
4172:′
4098:′
4012:∈
4001:for each
3989:∞
3983:‖
3968:‖
3960:∈
3897:′
3872:−
3856:′
3779:′
3754:−
3738:′
3725:σ
3655:′
3587:while if
3567:′
3507:while if
3408:′
3168:′
3160:⊆
3129:∈
3088:⊆
2965:∈
2911:A family
2894:′
2785:∞
2773:∈
2764:‖
2758:‖
2697:while if
2495:σ
2415:while if
2392:∈
2357:≤
2313:≤
2307:∘
2276:∈
2250:≤
2244:∘
2099:while if
2016:while if
1919:σ
1875:σ
1791:−
1778:∈
1771:⋂
1700:∈
1674:⊆
1639:⊆
1273:⊆
1238:∈
1212:⊆
1120:∈
1113:⋃
1029:∈
988:⊆
867:∈
838:→
807:A family
612:Fatou set
528:ϵ
52:sequences
48:countable
6570:(1966),
6554:(1987),
6542:21163277
6516:(1991).
6461:(1980),
6175:, p. 245
6040:See also
5974:. It is
5768: :
5718: :
5365:whereas
4912:for all
4711:. Since
4663:for all
4533:Suppose
4251:⟩
4230:⟨
3118:for all
2634:denotes
2437:barreled
2381:for all
2265:for all
2149:seminorm
1948:denotes
1227:for all
1018:for all
653:(i.e. a
588:Examples
543:for all
350:),
269:and all
250:),
197:and all
181:),
6452:, 2001
6187:, p. 29
5780:) <
5764:×
5739:, let
5714:×
5618:×
5464:. Then
5367:uniform
5339:
5335:arctan
5200:Hence,
4934:, then
4550:. Let
4321:Theorem
4137:barrels
3638:in the
377:,
298:,
212:,
125:be two
6634:
6624:
6604:
6594:
6540:
6530:
6502:
6492:
6473:
6405:
6233:
6225:
6208:
6200:
5997:, and
5899:A set
5802:> 0
5737:> 0
5730:) = 0}
5678:> 0
5638:(i.e.
5520:A set
5420:about
5385:A set
5371:filter
3529:normed
2296:Here,
395:, and
129:, and
6128:Notes
5579:)) â
5553:X Ă X
5247:} on
5234:with
4920:>
4836:>
4701:over
4530:Proof
4346:is a
4070:is a
3607:is a
3298:polar
2878:with
2852:be a
1377:into
781:into
680:into
655:group
571:When
6632:OCLC
6622:ISBN
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