5592:
4877:
1546:" actually represents a triple with the bonding maps and indexing set either defined elsewhere (e.g. canonical bonding maps, such as natural inclusions) or else the bonding maps are merely assumed to exist but there is no need to assign symbols to them (e.g. the bonding maps are not needed to state a theorem).
1318:
1429:
2121:
Direct limits in the categories of topological spaces, topological vector spaces (TVSs), and
Hausdorff locally convex TVSs are "poorly behaved". For instance, the direct limit of a sequence (i.e. indexed by the natural numbers) of locally convex
2772:
1174:
3522:
2353:
subset of itself. The strict inductive limit of a sequence of complete locally convex spaces (such as Fréchet spaces) is necessarily complete. In particular, every LF-space is complete. Every
2824:
2531:
3138:
3046:
2659:
3414:
2133:
to be
Hausdorff (in which case the direct limit does not exist in the category of Hausdorff TVSs). For this reason, only certain "well-behaved" direct systems are usually studied in
2936:
1327:
2965:
2688:
4913:
2563:
1461:
1051:
879:
793:
723:
690:
574:
428:
347:
158:
97:
4766:
3321:
3186:
2697:
2201:
3549:
3472:
3441:
3371:
2179:
1717:
1685:
1596:
2884:
237:
3081:
2992:
2851:
192:
4429:
1528:
is often omitted from the above tuple (i.e. not written); the same is true for the bonding maps if they are understood. Consequently, one often sees written "
1313:{\displaystyle X_{i}\xrightarrow {f_{i}^{j}} X_{j}\xrightarrow {f_{j}^{k}} X_{k}\;\;\;\;{\text{ is equal to }}\;\;\;\;X_{i}\xrightarrow {f_{i}^{k}} X_{k}.}
5040:
5015:
4592:
4997:
4719:
4574:
5465:
4967:
4906:
4550:
2051:
5034:
655:. If all objects in the category have an algebraic structure, then these maps are also assumed to be homomorphisms for that algebraic structure.
5210:
4066:. Grundlehren der mathematischen Wissenschaften. Vol. 159. Translated by Garling, D.J.H. New York: Springer Science & Business Media.
4390:
4360:
4326:
4292:
4262:
4235:
4205:
4171:
4109:
4071:
4041:
4006:
3957:
3923:
3896:
3808:
364:
If all objects in the category have an algebraic structure, then all morphisms are assumed to be homomorphisms for that algebraic structure.
5423:
5475:
4972:
4942:
5616:
5595:
4899:
4442:
4136:
3987:
5383:
4531:
4422:
3866:
3626:
3858:
1322:
If the above conditions are satisfied then the triple formed by the collections of these objects, morphisms, and the indexing set
4801:
4446:
854:
350:
3477:
5450:
5052:
5029:
4597:
4197:
4653:
5501:
4880:
4602:
4587:
4415:
4029:
3975:
2777:
2483:
2275:
In the category of locally convex topological vector spaces, the topology on a strict inductive limit of Fréchet spaces
1612:: In the case where the direct system is injective, it is often assumed without loss of generality that for all indices
4617:
3093:
3001:
2614:
5010:
5005:
4862:
4622:
4159:
5558:
5095:
4947:
4816:
4740:
4163:
3830:
3141:
2602:
4857:
5354:
5164:
4673:
4104:. Grundlehren der mathematischen Wissenschaften. Vol. 237. New York: Springer Science & Business Media.
3619:
Groups and geometric analysis : integral geometry, invariant differential operators, and spherical functions
3388:
2050:
Direct limits of directed direct systems always exist in the categories of sets, topological spaces, groups, and
4607:
5127:
5122:
5115:
5110:
4982:
4922:
4709:
4510:
1787:
745:
358:
164:
39:
5025:
4582:
5388:
5369:
5045:
4806:
3826:
2893:
2445:
2366:
2141:-spaces. However, non-Hausdorff locally convex inductive limits do occur in natural questions of analysis.
2086:
3335:
coincides with the topological inductive limit; that is, the direct limit of the finite dimensional spaces
5577:
5567:
5551:
5251:
5200:
5100:
5085:
4837:
4781:
4745:
3941:
2386:
1424:{\displaystyle \left(X_{\bullet },\left\{f_{i}^{j}\;:\;i,j\in I\;{\text{ and }}\;i\leq j\right\},I\right)}
354:
317:
1992:
of an injective directed inductive limit of locally convex spaces can be described by specifying that an
5546:
5246:
5233:
5215:
5180:
2338:
2082:
1993:
5020:
4280:
4352:
2941:
2664:
2369:. An LF-space that is the inductive limit of a countable sequence of separable spaces is separable.
5562:
5506:
5485:
4820:
4310:
3982:. Addison-Wesley series in mathematics. Vol. 1. Reading, MA: Addison-Wesley Publishing Company.
3378:
2767:{\displaystyle K_{1}\subset K_{2}\subset \ldots \subset K_{i}\subset \ldots \subset \mathbb {R} ^{n}}
2536:
1442:
1032:
860:
774:
704:
671:
555:
409:
328:
114:
53:
5445:
5440:
5398:
4977:
4786:
4724:
4438:
2374:
2134:
4344:
3304:
3169:
2184:
5430:
5373:
5322:
5318:
5307:
5292:
5288:
5159:
5149:
4811:
4678:
4189:
3527:
3450:
3419:
3349:
2378:
2362:
2330:
1599:
4097:
4059:
2152:
1690:
1658:
1569:
993:
belong to the original category (i.e. belong to the category of
Hausdorff topological spaces).
885:
In the category of topological spaces, the final topology always exists and moreover, a subset
5142:
5068:
4791:
4396:
4386:
4366:
4356:
4332:
4322:
4298:
4288:
4268:
4258:
4241:
4231:
4211:
4201:
4177:
4167:
4142:
4132:
4115:
4105:
4085:
4067:
4047:
4037:
4012:
4002:
3983:
3963:
3953:
3929:
3919:
3902:
3892:
3872:
3862:
3814:
3804:
3622:
3621:(Reprinted with corr. ed.). Providence, R.I: American Mathematical Society. p. 398.
1988:
In the category of locally convex topological vector spaces, the topology on the direct limit
1554:
For the construction of a direct limit of a general inductive system, please see the article:
2856:
5535:
5418:
5105:
5090:
4957:
4796:
4714:
4683:
4663:
4648:
4643:
4638:
4475:
3850:
2995:
2453:
2397:
2126:
497:
212:
100:
4081:
3059:
2970:
2829:
170:
5510:
5358:
4658:
4612:
4560:
4555:
4526:
4407:
4378:
4077:
4033:
3800:
3444:
3324:
2382:
2358:
2334:
969:
47:
4485:
2584:. Thus the strong dual space of an LF-space is a Fréchet space if and only if it is an
293:-space," so when reading mathematical literature, it is recommended to always check how
5541:
5490:
5205:
4847:
4699:
4500:
4223:
3946:
3884:
3570:
2070:
911:
661:
311:
161:
2181:
is an embedding of TVSs onto proper vector subspaces and if the system is directed by
5610:
5525:
5435:
5378:
5338:
5266:
5241:
5185:
5137:
5073:
4852:
4776:
4505:
4490:
4480:
2123:
2078:
2074:
2066:
5572:
5520:
5480:
5470:
5348:
5195:
5190:
4987:
4937:
4891:
4842:
4495:
4465:
4200:. Vol. 8 (Second ed.). New York, NY: Springer New York Imprint Springer.
3565:
1555:
1484:
1464:
1435:
1016:
1002:
373:
108:
3952:. Translated by Chaljub, Orlando. New York: Gordon and Breach Science Publishers.
2444:
A linear map from an LF-space into another TVS is continuous if and only if it is
5530:
5515:
5408:
5302:
5297:
5282:
5261:
5225:
5132:
4952:
4771:
4761:
4668:
4470:
3346:
in the category TOP and in the category TVS coincide. The continuous dual space
2470:
20:
5343:
5256:
5220:
5080:
4962:
4704:
4544:
4540:
4536:
4370:
2350:
4400:
4336:
4302:
4245:
4215:
4146:
4131:. Pure and applied mathematics (Second ed.). Boca Raton, FL: CRC Press.
4119:
4089:
3906:
3861:. Translated by Eggleston, H.G.; Madan, S. Berlin New York: Springer-Verlag.
3818:
5495:
5312:
4272:
3933:
3876:
4051:
4016:
3056:-space topology does not depend on the particular sequence of compact sets
4181:
3967:
3140:
is known as the space of test functions, of fundamental importance in the
5455:
5413:
5393:
5363:
5154:
4318:
3580:
3560:
2585:
2581:
2370:
2329:
An inductive limit in the category of locally convex TVSs of a family of
2215:. In such a situation we may assume without loss of generality that each
580:
is automatically assumed to have whatever algebraic structure is needed);
5403:
3575:
2694:-space structure is obtained by considering a sequence of compact sets
1883:'s have an algebraic structure, say addition for example, then for any
3327:. It is a LC topology, associated with the family of all seminorms on
2054:
TVSs. In the category of topological spaces, if every bonding map
1279:
1224:
1192:
3797:
Topological Vector Spaces: The Theory
Without Convexity Conditions
3301:, the latter space has the maximum among all TVS topologies on an
2365:, which together with completeness implies that every LF-space is
1922:
and then define their sum using by using the addition operator of
2473:
linear operator from an LF-space into another TVS is continuous.
3837:. Singapore-New Jersey-Hong Kong: Universitätsbibliothek: 35–133
3799:. Lecture Notes in Mathematics. Vol. 639. Berlin New York:
3702:
3700:
3600:
3598:
3596:
2279:
can be described by specifying that an absolutely convex subset
2203:
with its natural ordering, then the resulting limit is called a
4895:
4411:
3640:
3638:
3663:
3661:
3659:
3657:
3655:
3653:
3536:
3509:
3491:
3459:
3428:
3358:
2550:
1448:
1038:
866:
780:
710:
677:
561:
415:
334:
2396:
is the strict inductive limit of an increasing sequence of
2661:, the space of all infinitely differentiable functions on
1869:
is the final topology induced by these inclusion maps.
3530:
3517:{\displaystyle X_{\sigma }^{\prime }=X_{b}^{\prime }}
3480:
3453:
3422:
3391:
3352:
3307:
3172:
3096:
3062:
3004:
2973:
2944:
2896:
2859:
2832:
2780:
2700:
2667:
2617:
2539:
2486:
2187:
2155:
1693:
1661:
1572:
1445:
1330:
1177:
1035:
863:
777:
707:
674:
558:
412:
331:
215:
173:
117:
56:
4285:
Topological Vector Spaces, Distributions and
Kernels
3831:"An Introduction to Locally Convex Inductive Limits"
3795:
Adasch, Norbert; Ernst, Bruno; Keim, Dieter (1978).
2459:
is continuous if and only if its graph is closed in
500:" already has a well-known definition. The topology
5331:
5275:
5173:
5061:
4996:
4930:
4830:
4754:
4733:
4692:
4631:
4573:
4519:
4454:
251:. This means that the subspace topology induced on
4767:Spectral theory of ordinary differential equations
3945:
3543:
3516:
3466:
3435:
3408:
3365:
3315:
3180:
3132:
3075:
3040:
2986:
2959:
2930:
2878:
2845:
2819:{\displaystyle \bigcup _{i}K_{i}=\mathbb {R} ^{n}}
2818:
2766:
2682:
2653:
2557:
2526:{\displaystyle \left(X_{i}\right)_{i=1}^{\infty }}
2525:
2195:
2173:
1711:
1679:
1590:
1455:
1423:
1312:
1045:
873:
787:
717:
684:
568:
422:
341:
231:
186:
152:
91:
3718:
3691:
3385:, that is the space of all real valued sequences
3133:{\displaystyle C_{c}^{\infty }(\mathbb {R} ^{n})}
3041:{\displaystyle C_{c}^{\infty }(\mathbb {R} ^{n})}
2654:{\displaystyle C_{c}^{\infty }(\mathbb {R} ^{n})}
4154:Robertson, Alex P.; Robertson, Wendy J. (1980).
285:. Some authors (e.g. Schaefer) define the term "
4032:. Vol. 936. Berlin, Heidelberg, New York:
2886:. Such a sequence could be the balls of radius
1843:The limit maps are then the natural inclusions
4385:. Mineola, New York: Dover Publications, Inc.
4127:Narici, Lawrence; Beckenstein, Edward (2011).
4907:
4423:
3778:
3706:
3604:
3524:). In fact, it is the unique LC topology on
2597:Space of smooth compactly supported functions
8:
4026:Counterexamples in Topological Vector Spaces
3916:Functional Analysis: Theory and Applications
3679:
3331:. Also, the TVS inductive limit topology of
4383:Modern Methods in Topological Vector Spaces
3980:Topological Vector Spaces and Distributions
3409:{\displaystyle \mathbb {R} ^{\mathbb {N} }}
1790:than the original (i.e. given) topology on
4914:
4900:
4892:
4458:
4430:
4416:
4408:
4351:. Compact Textbooks in Mathematics. Cham:
4317:. Vol. 67. Amsterdam New York, N.Y.:
2938:of infinitely differentiable functions on
2238:and that the subspace topology induced on
1395:
1389:
1373:
1369:
1264:
1263:
1262:
1261:
1255:
1254:
1253:
1252:
3667:
3535:
3529:
3508:
3503:
3490:
3485:
3479:
3458:
3452:
3427:
3421:
3400:
3399:
3398:
3394:
3393:
3390:
3357:
3351:
3309:
3308:
3306:
3174:
3173:
3171:
3148:Direct limit of finite-dimensional spaces
3121:
3117:
3116:
3106:
3101:
3095:
3067:
3061:
3052:-space structure as described above. The
3029:
3025:
3024:
3014:
3009:
3003:
2978:
2972:
2951:
2947:
2946:
2943:
2919:
2906:
2901:
2895:
2864:
2858:
2837:
2831:
2810:
2806:
2805:
2795:
2785:
2779:
2758:
2754:
2753:
2737:
2718:
2705:
2699:
2674:
2670:
2669:
2666:
2642:
2638:
2637:
2627:
2622:
2616:
2549:
2544:
2538:
2517:
2506:
2496:
2485:
2261:is identical to the original topology on
2189:
2188:
2186:
2165:
2160:
2154:
1703:
1698:
1692:
1671:
1666:
1660:
1582:
1577:
1571:
1447:
1446:
1444:
1390:
1363:
1358:
1340:
1329:
1301:
1289:
1284:
1269:
1256:
1246:
1234:
1229:
1214:
1202:
1197:
1182:
1176:
1037:
1036:
1034:
865:
864:
862:
779:
778:
776:
709:
708:
706:
676:
675:
673:
610:is a family of maps where for each index
560:
559:
557:
414:
413:
411:
333:
332:
330:
274:is identical to the original topology on
220:
214:
178:
172:
138:
125:
116:
77:
64:
55:
4720:Group algebra of a locally compact group
4228:Handbook of Analysis and Its Foundations
3644:
3152:Suppose that for every positive integer
2302:is an absolutely convex neighborhood of
2023:is an absolutely convex neighborhood of
3855:Topological Vector Spaces: Chapters 1–5
3592:
3288:makes continuous the inclusions of the
1981:. This sum is independent of the index
458:is a topological space for every index
5053:Uniform boundedness (Banach–Steinhaus)
4255:An introduction to Functional Analysis
3766:
3754:
3742:
3730:
2931:{\displaystyle C_{c}^{\infty }(K_{i})}
2569:is a Fréchet space if and only if all
1169:where this means that the composition
496:'s "initial topology" since the term "
4349:A Course on Topological Vector Spaces
4287:. Mineola, N.Y.: Dover Publications.
2480:is an LF-space defined by a sequence
576:also have algebraic structures, then
306:Inductive/final/direct limit topology
7:
3835:Functional Analysis and Applications
3280:. Denote the resulting LF-space by
1029:there are (continuous) morphisms in
4162:. Vol. 53. Cambridge England:
3551:whose topological dual space is X.
2890:centered at the origin. The space
1764:) so that the subspace topology on
3107:
3015:
2967:with compact support contained in
2907:
2628:
2518:
1562:Direct limits of injective systems
1487:, the direct system is said to be
14:
4313:(1982). Nachbin, Leopoldo (ed.).
4230:. San Diego, CA: Academic Press.
2448:. A linear map from an LF-space
2422:if and only if there exists some
2341:) spaces has this same property.
239:is an embedding of TVSs then the
5591:
5590:
4876:
4875:
4802:Topological quantum field theory
3918:. New York: Dover Publications.
2960:{\displaystyle \mathbb {R} ^{n}}
2683:{\displaystyle \mathbb {R} ^{n}}
1865:. The direct limit topology on
1655:is identified with the range of
964:However, the final topology may
322:Throughout, it is assumed that
50:of a countable inductive system
5578:With the approximation property
4315:Topics in Locally Convex Spaces
4160:Cambridge Tracts in Mathematics
2853:is a subset of the interior of
2558:{\displaystyle X_{b}^{\prime }}
1550:Direct limit of a direct system
5041:Open mapping (Banach–Schauder)
3127:
3112:
3035:
3020:
2925:
2912:
2648:
2633:
1456:{\displaystyle {\mathcal {C}}}
1046:{\displaystyle {\mathcal {C}}}
874:{\displaystyle {\mathcal {C}}}
788:{\displaystyle {\mathcal {C}}}
718:{\displaystyle {\mathcal {C}}}
685:{\displaystyle {\mathcal {C}}}
569:{\displaystyle {\mathcal {C}}}
465:To avoid potential confusion,
423:{\displaystyle {\mathcal {C}}}
351:category of topological spaces
342:{\displaystyle {\mathcal {C}}}
153:{\displaystyle (X_{n},i_{nm})}
147:
118:
92:{\displaystyle (X_{n},i_{nm})}
86:
57:
1:
4598:Uniform boundedness principle
3719:Narici & Beckenstein 2011
3692:Narici & Beckenstein 2011
3323:-vector space with countable
194:is a Fréchet space. The name
4192:; Wolff, Manfred P. (1999).
4102:Topological Vector Spaces II
4030:Lecture Notes in Mathematics
3316:{\displaystyle \mathbb {R} }
3284:. Since any TVS topology on
3219:via the canonical embedding
3181:{\displaystyle \mathbb {R} }
2196:{\displaystyle \mathbb {N} }
2149:If each of the bonding maps
1970:is the addition operator of
1566:If each of the bonding maps
972:due to the requirement that
970:Hausdorff topological spaces
552:is a set (and if objects in
209:If each of the bonding maps
5262:Radially convex/Star-shaped
5247:Pre-compact/Totally bounded
4064:Topological Vector Spaces I
4024:Khaleelulla, S. M. (1982).
4001:. Stuttgart: B.G. Teubner.
3914:Edwards, Robert E. (1995).
3891:. Boston: Allyn and Bacon.
3617:Helgason, Sigurdur (2000).
3544:{\displaystyle X^{\prime }}
3467:{\displaystyle X^{\prime }}
3436:{\displaystyle X^{\prime }}
3366:{\displaystyle X^{\prime }}
2533:then the strong dual space
2377:and their strong duals are
1687:) and that the bonding map
1537:is a direct system" where "
353:or some subcategory of the
5633:
4948:Continuous linear operator
4741:Invariant subspace problem
4164:Cambridge University Press
2690:with compact support. The
2603:Distribution (mathematics)
2600:
2117:Problem with direct limits
1719:is the natural inclusion
1602:then the system is called
1000:
929:is open (resp. closed) in
895:is open (resp. closed) in
406:is a family of objects in
315:
309:
5617:Topological vector spaces
5586:
5293:Algebraic interior (core)
5035:Vector-valued Hahn–Banach
4923:Topological vector spaces
4871:
4461:
4194:Topological Vector Spaces
4156:Topological Vector Spaces
4129:Topological Vector Spaces
3948:Topological Vector Spaces
3779:Schaefer & Wolff 1999
3707:Schaefer & Wolff 1999
3605:Schaefer & Wolff 1999
3416:and the weak topology on
2174:{\displaystyle f_{i}^{j}}
1712:{\displaystyle f_{i}^{j}}
1680:{\displaystyle f_{i}^{j}}
1591:{\displaystyle f_{i}^{j}}
1479:. Since the indexing set
1019:and that for all indices
968:exist in the category of
614:, the map has prototype
359:topological vector spaces
165:topological vector spaces
46:that is a locally convex
5123:Topological homomorphism
4983:Topological vector space
4710:Spectrum of a C*-algebra
4253:Swartz, Charles (1992).
3859:Éléments de mathématique
3208:as a vector subspace of
2607:A typical example of an
2226:is a vector subspace of
1804:In this case, also take
1633:is a vector subspace of
289:-space" to mean "strict
40:topological vector space
16:Topological vector space
4807:Noncommutative geometry
4257:. New York: M. Dekker.
3942:Grothendieck, Alexander
3827:Bierstedt, Klaus-Dieter
3142:theory of distributions
2879:{\displaystyle K_{i+1}}
2446:sequentially continuous
2430:is a bounded subset of
2137:. Such systems include
1258: is equal to
1133:compatibility condition
1106:is the identity map on
659:If it exists, then the
5181:Absolutely convex/disk
4863:Tomita–Takesaki theory
4838:Approximation property
4782:Calculus of variations
3997:Jarchow, Hans (1981).
3545:
3518:
3468:
3437:
3410:
3367:
3317:
3182:
3134:
3077:
3042:
2988:
2961:
2932:
2880:
2847:
2820:
2768:
2684:
2655:
2559:
2527:
2387:Alexander Grothendieck
2197:
2175:
2145:Strict inductive limit
1713:
1681:
1592:
1457:
1425:
1314:
1047:
875:
789:
719:
686:
570:
424:
343:
318:Category (mathematics)
233:
232:{\displaystyle i_{nm}}
188:
154:
93:
5216:Complemented subspace
5030:hyperplane separation
4858:Banach–Mazur distance
4821:Generalized functions
3999:Locally convex spaces
3546:
3519:
3469:
3438:
3411:
3368:
3318:
3183:
3135:
3078:
3076:{\displaystyle K_{i}}
3043:
2989:
2987:{\displaystyle K_{i}}
2962:
2933:
2881:
2848:
2846:{\displaystyle K_{i}}
2821:
2769:
2685:
2656:
2560:
2528:
2283:is a neighborhood of
2198:
2176:
2083:topological embedding
2004:is a neighborhood of
1788:weaker (i.e. coarser)
1714:
1682:
1593:
1458:
1426:
1315:
1048:
876:
790:
720:
687:
571:
425:
344:
234:
189:
187:{\displaystyle X_{n}}
155:
94:
5466:Locally convex space
5016:Closed graph theorem
4968:Locally convex space
4603:Kakutani fixed-point
4588:Riesz representation
3528:
3478:
3451:
3420:
3389:
3379:algebraic dual space
3350:
3305:
3170:
3094:
3060:
3002:
2971:
2942:
2894:
2857:
2830:
2778:
2698:
2665:
2615:
2537:
2484:
2349:Every LF-space is a
2185:
2153:
2046:Direct limits in Top
1897:, we pick any index
1691:
1659:
1570:
1520:If the indexing set
1443:
1328:
1175:
1033:
861:
775:
705:
672:
556:
410:
329:
213:
171:
115:
103:. This means that
54:
5446:Interpolation space
4978:Operator topologies
4787:Functional calculus
4746:Mahler's conjecture
4725:Von Neumann algebra
4439:Functional analysis
4190:Schaefer, Helmut H.
3682:, pp. 130–142.
3647:, pp. 420–435.
3513:
3495:
3111:
3019:
2911:
2632:
2554:
2522:
2170:
2135:functional analysis
2089:) then so is every
1708:
1676:
1587:
1524:is understood then
1368:
1295:
1294:
1240:
1239:
1208:
1207:
1131:then the following
297:-space is defined.
243:-space is called a
160:in the category of
111:of a direct system
5476:(Pseudo)Metrizable
5308:Minkowski addition
5160:Sublinear function
4812:Riemann hypothesis
4511:Topological vector
3541:
3514:
3499:
3481:
3464:
3433:
3406:
3363:
3313:
3178:
3130:
3097:
3090:-space structure,
3073:
3038:
3005:
2984:
2957:
2928:
2897:
2876:
2843:
2816:
2790:
2764:
2680:
2651:
2618:
2555:
2540:
2523:
2487:
2193:
2171:
2156:
2132:
1709:
1694:
1677:
1662:
1588:
1573:
1453:
1421:
1354:
1310:
1280:
1225:
1193:
1043:
871:
785:
715:
699:inductive topology
693:, also called the
682:
566:
420:
339:
229:
184:
150:
89:
5604:
5603:
5323:Relative interior
5069:Bilinear operator
4953:Linear functional
4889:
4888:
4792:Integral operator
4569:
4568:
4392:978-0-486-49353-4
4362:978-3-030-32945-7
4328:978-0-08-087178-3
4321:Science Pub. Co.
4294:978-0-486-45352-1
4264:978-0-8247-8643-4
4237:978-0-12-622760-4
4207:978-1-4612-7155-0
4173:978-0-521-29882-7
4111:978-0-387-90400-9
4073:978-3-642-64988-2
4043:978-3-540-11565-6
4008:978-3-519-02224-4
3959:978-0-677-30020-7
3925:978-0-486-68143-6
3898:978-0-697-06889-7
3851:Bourbaki, Nicolas
3810:978-3-540-08662-8
3709:, pp. 59–61.
3680:Grothendieck 1973
3670:, pp. 41–56.
3607:, pp. 55–61.
2781:
2385:(a result due to
2367:ultrabornological
2130:
1994:absolutely convex
1754:(i.e. defined by
1393:
1296:
1259:
1241:
1209:
725:, and denoted by
5624:
5594:
5593:
5568:Uniformly smooth
5237:
5229:
5196:Balanced/Circled
5186:Absorbing/Radial
4916:
4909:
4902:
4893:
4879:
4878:
4797:Jones polynomial
4715:Operator algebra
4459:
4432:
4425:
4418:
4409:
4404:
4379:Wilansky, Albert
4374:
4353:Birkhäuser Basel
4340:
4311:Valdivia, Manuel
4306:
4281:Trèves, François
4276:
4249:
4219:
4185:
4150:
4123:
4098:Köthe, Gottfried
4093:
4060:Köthe, Gottfried
4055:
4020:
3993:
3971:
3951:
3937:
3910:
3880:
3846:
3844:
3842:
3822:
3782:
3776:
3770:
3764:
3758:
3752:
3746:
3740:
3734:
3728:
3722:
3716:
3710:
3704:
3695:
3689:
3683:
3677:
3671:
3665:
3648:
3642:
3633:
3632:
3614:
3608:
3602:
3550:
3548:
3547:
3542:
3540:
3539:
3523:
3521:
3520:
3515:
3512:
3507:
3494:
3489:
3473:
3471:
3470:
3465:
3463:
3462:
3443:is equal to the
3442:
3440:
3439:
3434:
3432:
3431:
3415:
3413:
3412:
3407:
3405:
3404:
3403:
3397:
3384:
3377:is equal to the
3376:
3372:
3370:
3369:
3364:
3362:
3361:
3345:
3334:
3330:
3322:
3320:
3319:
3314:
3312:
3300:
3287:
3283:
3279:
3278:
3238:
3218:
3198:
3188:
3187:
3185:
3184:
3179:
3177:
3155:
3139:
3137:
3136:
3131:
3126:
3125:
3120:
3110:
3105:
3082:
3080:
3079:
3074:
3072:
3071:
3047:
3045:
3044:
3039:
3034:
3033:
3028:
3018:
3013:
2993:
2991:
2990:
2985:
2983:
2982:
2966:
2964:
2963:
2958:
2956:
2955:
2950:
2937:
2935:
2934:
2929:
2924:
2923:
2910:
2905:
2885:
2883:
2882:
2877:
2875:
2874:
2852:
2850:
2849:
2844:
2842:
2841:
2826:and for all i,
2825:
2823:
2822:
2817:
2815:
2814:
2809:
2800:
2799:
2789:
2773:
2771:
2770:
2765:
2763:
2762:
2757:
2742:
2741:
2723:
2722:
2710:
2709:
2689:
2687:
2686:
2681:
2679:
2678:
2673:
2660:
2658:
2657:
2652:
2647:
2646:
2641:
2631:
2626:
2579:
2568:
2564:
2562:
2561:
2556:
2553:
2548:
2532:
2530:
2529:
2524:
2521:
2516:
2505:
2501:
2500:
2479:
2468:
2458:
2451:
2440:
2429:
2425:
2421:
2417:
2413:
2409:
2395:
2320:
2316:
2305:
2301:
2286:
2282:
2278:
2271:
2260:
2248:
2237:
2225:
2202:
2200:
2199:
2194:
2192:
2180:
2178:
2177:
2172:
2169:
2164:
2112:
2064:
2041:
2038:for every index
2037:
2026:
2022:
2007:
2003:
1999:
1991:
1985:that is chosen.
1984:
1980:
1969:
1957:
1932:
1921:
1902:
1896:
1882:
1868:
1864:
1840:
1831:
1830:
1829:
1828:
1818:
1817:
1800:
1785:
1774:
1763:
1751:
1732:
1731:
1718:
1716:
1715:
1710:
1707:
1702:
1686:
1684:
1683:
1678:
1675:
1670:
1654:
1644:(in particular,
1643:
1632:
1621:
1597:
1595:
1594:
1589:
1586:
1581:
1545:
1536:
1527:
1523:
1501:
1482:
1478:
1462:
1460:
1459:
1454:
1452:
1451:
1439:in the category
1430:
1428:
1427:
1422:
1420:
1416:
1409:
1405:
1394:
1391:
1367:
1362:
1345:
1344:
1319:
1317:
1316:
1311:
1306:
1305:
1293:
1288:
1275:
1274:
1273:
1260:
1257:
1251:
1250:
1238:
1233:
1220:
1219:
1218:
1206:
1201:
1188:
1187:
1186:
1165:
1130:
1116:
1105:
1094:
1082:
1052:
1050:
1049:
1044:
1042:
1041:
1028:
1014:
992:
991:
976:
960:
957:for every index
956:
955:
933:
928:
910:
909:
899:
894:
880:
878:
877:
872:
870:
869:
852:
851:
841:
837:
815:
801:
798:for every index
794:
792:
791:
786:
784:
783:
771:is an object in
770:
769:
759:
751:
743:
736:
724:
722:
721:
716:
714:
713:
691:
689:
688:
683:
681:
680:
667:
654:
649:
627:
613:
609:
579:
575:
573:
572:
567:
565:
564:
551:
539:
528:
513:
498:initial topology
495:
478:
461:
457:
456:
434:
429:
427:
426:
421:
419:
418:
405:
371:
348:
346:
345:
340:
338:
337:
284:
273:
261:
238:
236:
235:
230:
228:
227:
193:
191:
190:
185:
183:
182:
159:
157:
156:
151:
146:
145:
130:
129:
98:
96:
95:
90:
85:
84:
69:
68:
5632:
5631:
5627:
5626:
5625:
5623:
5622:
5621:
5607:
5606:
5605:
5600:
5582:
5344:B-complete/Ptak
5327:
5271:
5235:
5227:
5206:Bounding points
5169:
5111:Densely defined
5057:
5046:Bounded inverse
4992:
4926:
4920:
4890:
4885:
4867:
4831:Advanced topics
4826:
4750:
4729:
4688:
4654:Hilbert–Schmidt
4627:
4618:Gelfand–Naimark
4565:
4515:
4450:
4436:
4393:
4377:
4363:
4343:
4329:
4309:
4295:
4279:
4265:
4252:
4238:
4224:Schechter, Eric
4222:
4208:
4188:
4174:
4153:
4139:
4126:
4112:
4096:
4074:
4058:
4044:
4034:Springer-Verlag
4023:
4009:
3996:
3990:
3974:
3960:
3940:
3926:
3913:
3899:
3885:Dugundji, James
3883:
3869:
3849:
3840:
3838:
3825:
3811:
3801:Springer-Verlag
3794:
3791:
3786:
3785:
3777:
3773:
3765:
3761:
3753:
3749:
3741:
3737:
3729:
3725:
3717:
3713:
3705:
3698:
3690:
3686:
3678:
3674:
3666:
3651:
3643:
3636:
3629:
3616:
3615:
3611:
3603:
3594:
3589:
3557:
3531:
3526:
3525:
3476:
3475:
3454:
3449:
3448:
3445:strong topology
3423:
3418:
3417:
3392:
3387:
3386:
3382:
3374:
3353:
3348:
3347:
3344:
3336:
3332:
3328:
3325:Hamel dimension
3303:
3302:
3298:
3296:
3285:
3281:
3276:
3267:
3260:
3251:
3241:
3240:
3237:
3228:
3220:
3217:
3209:
3207:
3190:
3168:
3167:
3165:
3157:
3153:
3150:
3115:
3092:
3091:
3063:
3058:
3057:
3023:
3000:
2999:
2998:structure and
2974:
2969:
2968:
2945:
2940:
2939:
2915:
2892:
2891:
2860:
2855:
2854:
2833:
2828:
2827:
2804:
2791:
2776:
2775:
2752:
2733:
2714:
2701:
2696:
2695:
2668:
2663:
2662:
2636:
2613:
2612:
2605:
2599:
2594:
2578:
2570:
2566:
2535:
2534:
2492:
2488:
2482:
2481:
2477:
2460:
2456:
2449:
2439:
2431:
2427:
2423:
2419:
2415:
2411:
2408:
2400:
2393:
2347:
2339:quasi-barrelled
2327:
2318:
2315:
2307:
2303:
2300:
2288:
2287:if and only if
2284:
2280:
2276:
2270:
2262:
2259:
2250:
2247:
2239:
2236:
2227:
2224:
2216:
2183:
2182:
2151:
2150:
2147:
2119:
2107:
2098:
2090:
2063:
2055:
2039:
2036:
2028:
2024:
2021:
2009:
2008:if and only if
2005:
2001:
1997:
1989:
1982:
1979:
1971:
1968:
1962:
1959:
1953:
1936:
1931:
1923:
1920:
1904:
1898:
1884:
1881:
1873:
1870:
1866:
1859:
1850:
1844:
1842:
1839:
1820:
1819:
1815:
1814:
1813:
1812:
1807:
1799:
1791:
1784:
1776:
1773:
1765:
1755:
1752:
1750:
1741:
1730:
1725:
1724:
1723:
1721:
1689:
1688:
1657:
1656:
1653:
1645:
1642:
1634:
1631:
1623:
1613:
1568:
1567:
1552:
1544:
1538:
1535:
1529:
1525:
1521:
1517:of the system.
1502:are called the
1500:
1492:
1480:
1474:
1441:
1440:
1431:
1392: and
1353:
1349:
1336:
1335:
1331:
1326:
1325:
1320:
1297:
1265:
1242:
1210:
1178:
1173:
1172:
1167:
1164:
1155:
1146:
1138:
1118:
1115:
1107:
1104:
1096:
1086:
1083:
1081:
1072:
1063:
1055:
1031:
1030:
1020:
1008:
1005:
999:
989:
988:
987:
974:
973:
958:
953:
952:
951:
941:
931:
930:
922:
914:
907:
906:
897:
896:
886:
859:
858:
849:
848:
839:
835:
834:
833:
823:
813:
811:
803:
799:
773:
772:
767:
766:
757:
756:
749:
746:finest topology
742:
738:
735:
734:
726:
703:
702:
670:
669:
665:
647:
646:
645:
635:
625:
623:
615:
611:
608:
598:
589:
583:
577:
554:
553:
549:
538:
530:
527:
519:
512:
511:
501:
494:
486:
477:
476:
466:
459:
454:
453:
452:
442:
432:
431:
408:
407:
404:
394:
385:
379:
372:is a non-empty
369:
327:
326:
320:
314:
308:
303:
283:
275:
272:
263:
260:
252:
216:
211:
210:
206:réchet spaces.
174:
169:
168:
134:
121:
113:
112:
73:
60:
52:
51:
48:inductive limit
30:, also written
17:
12:
11:
5:
5630:
5628:
5620:
5619:
5609:
5608:
5602:
5601:
5599:
5598:
5587:
5584:
5583:
5581:
5580:
5575:
5570:
5565:
5563:Ultrabarrelled
5555:
5549:
5544:
5538:
5533:
5528:
5523:
5518:
5513:
5504:
5498:
5493:
5491:Quasi-complete
5488:
5486:Quasibarrelled
5483:
5478:
5473:
5468:
5463:
5458:
5453:
5448:
5443:
5438:
5433:
5428:
5427:
5426:
5416:
5411:
5406:
5401:
5396:
5391:
5386:
5381:
5376:
5366:
5361:
5351:
5346:
5341:
5335:
5333:
5329:
5328:
5326:
5325:
5315:
5310:
5305:
5300:
5295:
5285:
5279:
5277:
5276:Set operations
5273:
5272:
5270:
5269:
5264:
5259:
5254:
5249:
5244:
5239:
5231:
5223:
5218:
5213:
5208:
5203:
5198:
5193:
5188:
5183:
5177:
5175:
5171:
5170:
5168:
5167:
5162:
5157:
5152:
5147:
5146:
5145:
5140:
5135:
5125:
5120:
5119:
5118:
5113:
5108:
5103:
5098:
5093:
5088:
5078:
5077:
5076:
5065:
5063:
5059:
5058:
5056:
5055:
5050:
5049:
5048:
5038:
5032:
5023:
5018:
5013:
5011:Banach–Alaoglu
5008:
5006:Anderson–Kadec
5002:
5000:
4994:
4993:
4991:
4990:
4985:
4980:
4975:
4970:
4965:
4960:
4955:
4950:
4945:
4940:
4934:
4932:
4931:Basic concepts
4928:
4927:
4921:
4919:
4918:
4911:
4904:
4896:
4887:
4886:
4884:
4883:
4872:
4869:
4868:
4866:
4865:
4860:
4855:
4850:
4848:Choquet theory
4845:
4840:
4834:
4832:
4828:
4827:
4825:
4824:
4814:
4809:
4804:
4799:
4794:
4789:
4784:
4779:
4774:
4769:
4764:
4758:
4756:
4752:
4751:
4749:
4748:
4743:
4737:
4735:
4731:
4730:
4728:
4727:
4722:
4717:
4712:
4707:
4702:
4700:Banach algebra
4696:
4694:
4690:
4689:
4687:
4686:
4681:
4676:
4671:
4666:
4661:
4656:
4651:
4646:
4641:
4635:
4633:
4629:
4628:
4626:
4625:
4623:Banach–Alaoglu
4620:
4615:
4610:
4605:
4600:
4595:
4590:
4585:
4579:
4577:
4571:
4570:
4567:
4566:
4564:
4563:
4558:
4553:
4551:Locally convex
4548:
4534:
4529:
4523:
4521:
4517:
4516:
4514:
4513:
4508:
4503:
4498:
4493:
4488:
4483:
4478:
4473:
4468:
4462:
4456:
4452:
4451:
4437:
4435:
4434:
4427:
4420:
4412:
4406:
4405:
4391:
4375:
4361:
4341:
4327:
4307:
4293:
4277:
4263:
4250:
4236:
4220:
4206:
4186:
4172:
4151:
4138:978-1584888666
4137:
4124:
4110:
4094:
4072:
4056:
4042:
4021:
4007:
3994:
3989:978-0201029857
3988:
3972:
3958:
3938:
3924:
3911:
3897:
3881:
3867:
3847:
3823:
3809:
3790:
3787:
3784:
3783:
3781:, p. 201.
3771:
3769:, p. 201.
3759:
3757:, p. 142.
3747:
3745:, p. 173.
3735:
3733:, p. 141.
3723:
3721:, p. 436.
3711:
3696:
3694:, p. 435.
3684:
3672:
3668:Bierstedt 1988
3649:
3634:
3627:
3609:
3591:
3590:
3588:
3585:
3584:
3583:
3578:
3573:
3571:Final topology
3568:
3563:
3556:
3553:
3538:
3534:
3511:
3506:
3502:
3498:
3493:
3488:
3484:
3461:
3457:
3430:
3426:
3402:
3396:
3360:
3356:
3340:
3311:
3292:
3272:
3265:
3256:
3249:
3233:
3224:
3213:
3203:
3176:
3161:
3149:
3146:
3129:
3124:
3119:
3114:
3109:
3104:
3100:
3070:
3066:
3037:
3032:
3027:
3022:
3017:
3012:
3008:
2994:has a natural
2981:
2977:
2954:
2949:
2927:
2922:
2918:
2914:
2909:
2904:
2900:
2873:
2870:
2867:
2863:
2840:
2836:
2813:
2808:
2803:
2798:
2794:
2788:
2784:
2761:
2756:
2751:
2748:
2745:
2740:
2736:
2732:
2729:
2726:
2721:
2717:
2713:
2708:
2704:
2677:
2672:
2650:
2645:
2640:
2635:
2630:
2625:
2621:
2601:Main article:
2598:
2595:
2593:
2590:
2574:
2552:
2547:
2543:
2520:
2515:
2512:
2509:
2504:
2499:
2495:
2491:
2435:
2418:is bounded in
2410:then a subset
2404:
2346:
2343:
2326:
2323:
2311:
2296:
2266:
2254:
2243:
2231:
2220:
2191:
2168:
2163:
2159:
2146:
2143:
2127:Fréchet spaces
2118:
2115:
2103:
2094:
2059:
2052:locally convex
2032:
2017:
1975:
1964:
1949:
1935:
1927:
1916:
1877:
1855:
1846:
1835:
1806:
1795:
1780:
1769:
1746:
1737:
1726:
1720:
1706:
1701:
1697:
1674:
1669:
1665:
1649:
1638:
1627:
1608:
1585:
1580:
1576:
1551:
1548:
1542:
1533:
1496:
1450:
1433:is known as a
1419:
1415:
1412:
1408:
1404:
1401:
1398:
1388:
1385:
1382:
1379:
1376:
1372:
1366:
1361:
1357:
1352:
1348:
1343:
1339:
1334:
1324:
1309:
1304:
1300:
1292:
1287:
1283:
1278:
1272:
1268:
1249:
1245:
1237:
1232:
1228:
1223:
1217:
1213:
1205:
1200:
1196:
1191:
1185:
1181:
1171:
1160:
1151:
1142:
1137:
1135:is satisfied:
1111:
1100:
1077:
1068:
1059:
1054:
1040:
1001:Main article:
998:
997:Direct systems
995:
985:
981:
967:
947:
943:
937:
918:
912:if and only if
904:
883:
882:
868:
846:
829:
825:
819:
807:
796:
782:
764:
740:
732:
728:
712:
679:
662:final topology
657:
656:
641:
637:
631:
619:
600:
594:
587:
581:
563:
547:
546:
545:
542:given topology
534:
523:
514:is called the
507:
503:
490:
472:
468:
448:
444:
438:
417:
396:
390:
383:
377:
367:
366:
365:
349:is either the
336:
312:Final topology
310:Main article:
307:
304:
302:
299:
279:
267:
256:
226:
223:
219:
181:
177:
162:locally convex
149:
144:
141:
137:
133:
128:
124:
120:
101:Fréchet spaces
88:
83:
80:
76:
72:
67:
63:
59:
15:
13:
10:
9:
6:
4:
3:
2:
5629:
5618:
5615:
5614:
5612:
5597:
5589:
5588:
5585:
5579:
5576:
5574:
5571:
5569:
5566:
5564:
5560:
5556:
5554:) convex
5553:
5550:
5548:
5545:
5543:
5539:
5537:
5534:
5532:
5529:
5527:
5526:Semi-complete
5524:
5522:
5519:
5517:
5514:
5512:
5508:
5505:
5503:
5499:
5497:
5494:
5492:
5489:
5487:
5484:
5482:
5479:
5477:
5474:
5472:
5469:
5467:
5464:
5462:
5459:
5457:
5454:
5452:
5449:
5447:
5444:
5442:
5441:Infrabarreled
5439:
5437:
5434:
5432:
5429:
5425:
5422:
5421:
5420:
5417:
5415:
5412:
5410:
5407:
5405:
5402:
5400:
5399:Distinguished
5397:
5395:
5392:
5390:
5387:
5385:
5382:
5380:
5377:
5375:
5371:
5367:
5365:
5362:
5360:
5356:
5352:
5350:
5347:
5345:
5342:
5340:
5337:
5336:
5334:
5332:Types of TVSs
5330:
5324:
5320:
5316:
5314:
5311:
5309:
5306:
5304:
5301:
5299:
5296:
5294:
5290:
5286:
5284:
5281:
5280:
5278:
5274:
5268:
5265:
5263:
5260:
5258:
5255:
5253:
5252:Prevalent/Shy
5250:
5248:
5245:
5243:
5242:Extreme point
5240:
5238:
5232:
5230:
5224:
5222:
5219:
5217:
5214:
5212:
5209:
5207:
5204:
5202:
5199:
5197:
5194:
5192:
5189:
5187:
5184:
5182:
5179:
5178:
5176:
5174:Types of sets
5172:
5166:
5163:
5161:
5158:
5156:
5153:
5151:
5148:
5144:
5141:
5139:
5136:
5134:
5131:
5130:
5129:
5126:
5124:
5121:
5117:
5116:Discontinuous
5114:
5112:
5109:
5107:
5104:
5102:
5099:
5097:
5094:
5092:
5089:
5087:
5084:
5083:
5082:
5079:
5075:
5072:
5071:
5070:
5067:
5066:
5064:
5060:
5054:
5051:
5047:
5044:
5043:
5042:
5039:
5036:
5033:
5031:
5027:
5024:
5022:
5019:
5017:
5014:
5012:
5009:
5007:
5004:
5003:
5001:
4999:
4995:
4989:
4986:
4984:
4981:
4979:
4976:
4974:
4973:Metrizability
4971:
4969:
4966:
4964:
4961:
4959:
4958:Fréchet space
4956:
4954:
4951:
4949:
4946:
4944:
4941:
4939:
4936:
4935:
4933:
4929:
4924:
4917:
4912:
4910:
4905:
4903:
4898:
4897:
4894:
4882:
4874:
4873:
4870:
4864:
4861:
4859:
4856:
4854:
4853:Weak topology
4851:
4849:
4846:
4844:
4841:
4839:
4836:
4835:
4833:
4829:
4822:
4818:
4815:
4813:
4810:
4808:
4805:
4803:
4800:
4798:
4795:
4793:
4790:
4788:
4785:
4783:
4780:
4778:
4777:Index theorem
4775:
4773:
4770:
4768:
4765:
4763:
4760:
4759:
4757:
4753:
4747:
4744:
4742:
4739:
4738:
4736:
4734:Open problems
4732:
4726:
4723:
4721:
4718:
4716:
4713:
4711:
4708:
4706:
4703:
4701:
4698:
4697:
4695:
4691:
4685:
4682:
4680:
4677:
4675:
4672:
4670:
4667:
4665:
4662:
4660:
4657:
4655:
4652:
4650:
4647:
4645:
4642:
4640:
4637:
4636:
4634:
4630:
4624:
4621:
4619:
4616:
4614:
4611:
4609:
4606:
4604:
4601:
4599:
4596:
4594:
4591:
4589:
4586:
4584:
4581:
4580:
4578:
4576:
4572:
4562:
4559:
4557:
4554:
4552:
4549:
4546:
4542:
4538:
4535:
4533:
4530:
4528:
4525:
4524:
4522:
4518:
4512:
4509:
4507:
4504:
4502:
4499:
4497:
4494:
4492:
4489:
4487:
4484:
4482:
4479:
4477:
4474:
4472:
4469:
4467:
4464:
4463:
4460:
4457:
4453:
4448:
4444:
4440:
4433:
4428:
4426:
4421:
4419:
4414:
4413:
4410:
4402:
4398:
4394:
4388:
4384:
4380:
4376:
4372:
4368:
4364:
4358:
4354:
4350:
4346:
4345:Voigt, Jürgen
4342:
4338:
4334:
4330:
4324:
4320:
4316:
4312:
4308:
4304:
4300:
4296:
4290:
4286:
4282:
4278:
4274:
4270:
4266:
4260:
4256:
4251:
4247:
4243:
4239:
4233:
4229:
4225:
4221:
4217:
4213:
4209:
4203:
4199:
4195:
4191:
4187:
4183:
4179:
4175:
4169:
4165:
4161:
4157:
4152:
4148:
4144:
4140:
4134:
4130:
4125:
4121:
4117:
4113:
4107:
4103:
4099:
4095:
4091:
4087:
4083:
4079:
4075:
4069:
4065:
4061:
4057:
4053:
4049:
4045:
4039:
4035:
4031:
4027:
4022:
4018:
4014:
4010:
4004:
4000:
3995:
3991:
3985:
3981:
3977:
3976:Horváth, John
3973:
3969:
3965:
3961:
3955:
3950:
3949:
3943:
3939:
3935:
3931:
3927:
3921:
3917:
3912:
3908:
3904:
3900:
3894:
3890:
3886:
3882:
3878:
3874:
3870:
3868:3-540-13627-4
3864:
3860:
3856:
3852:
3848:
3836:
3832:
3828:
3824:
3820:
3816:
3812:
3806:
3802:
3798:
3793:
3792:
3788:
3780:
3775:
3772:
3768:
3763:
3760:
3756:
3751:
3748:
3744:
3739:
3736:
3732:
3727:
3724:
3720:
3715:
3712:
3708:
3703:
3701:
3697:
3693:
3688:
3685:
3681:
3676:
3673:
3669:
3664:
3662:
3660:
3658:
3656:
3654:
3650:
3646:
3645:Dugundji 1966
3641:
3639:
3635:
3630:
3628:0-8218-2673-5
3624:
3620:
3613:
3610:
3606:
3601:
3599:
3597:
3593:
3586:
3582:
3579:
3577:
3574:
3572:
3569:
3567:
3564:
3562:
3559:
3558:
3554:
3552:
3532:
3504:
3500:
3496:
3486:
3482:
3455:
3446:
3424:
3380:
3354:
3343:
3339:
3326:
3295:
3291:
3275:
3271:
3264:
3259:
3255:
3248:
3244:
3236:
3232:
3227:
3223:
3216:
3212:
3206:
3202:
3197:
3193:
3164:
3160:
3147:
3145:
3143:
3122:
3102:
3098:
3089:
3084:
3068:
3064:
3055:
3051:
3048:inherits its
3030:
3010:
3006:
2997:
2996:Fréchet space
2979:
2975:
2952:
2920:
2916:
2902:
2898:
2889:
2871:
2868:
2865:
2861:
2838:
2834:
2811:
2801:
2796:
2792:
2786:
2782:
2759:
2749:
2746:
2743:
2738:
2734:
2730:
2727:
2724:
2719:
2715:
2711:
2706:
2702:
2693:
2675:
2643:
2623:
2619:
2610:
2604:
2596:
2591:
2589:
2587:
2583:
2577:
2573:
2545:
2541:
2513:
2510:
2507:
2502:
2497:
2493:
2489:
2474:
2472:
2467:
2463:
2455:
2454:Fréchet space
2447:
2442:
2438:
2434:
2407:
2403:
2399:
2398:Fréchet space
2390:
2388:
2384:
2380:
2376:
2375:distinguished
2372:
2368:
2364:
2360:
2356:
2352:
2344:
2342:
2340:
2336:
2332:
2324:
2322:
2314:
2310:
2299:
2295:
2291:
2273:
2269:
2265:
2257:
2253:
2246:
2242:
2234:
2230:
2223:
2219:
2214:
2210:
2206:
2166:
2161:
2157:
2144:
2142:
2140:
2136:
2128:
2125:
2116:
2114:
2111:
2106:
2102:
2097:
2093:
2088:
2084:
2080:
2079:homeomorphism
2076:
2072:
2068:
2062:
2058:
2053:
2048:
2047:
2043:
2035:
2031:
2020:
2016:
2012:
1995:
1986:
1978:
1974:
1967:
1956:
1952:
1947:
1943:
1939:
1934:
1930:
1926:
1919:
1915:
1911:
1907:
1901:
1895:
1891:
1887:
1880:
1876:
1863:
1858:
1854:
1849:
1838:
1834:
1827:
1823:
1810:
1805:
1802:
1798:
1794:
1789:
1783:
1779:
1772:
1768:
1762:
1758:
1749:
1745:
1740:
1736:
1729:
1704:
1699:
1695:
1672:
1667:
1663:
1652:
1648:
1641:
1637:
1630:
1626:
1620:
1616:
1611:
1607:
1605:
1601:
1583:
1578:
1574:
1564:
1563:
1559:
1557:
1549:
1547:
1541:
1532:
1518:
1516:
1513:
1509:
1505:
1499:
1495:
1490:
1486:
1477:
1472:
1468:
1467:
1438:
1437:
1436:direct system
1417:
1413:
1410:
1406:
1402:
1399:
1396:
1386:
1383:
1380:
1377:
1374:
1370:
1364:
1359:
1355:
1350:
1346:
1341:
1337:
1332:
1323:
1307:
1302:
1298:
1290:
1285:
1281:
1276:
1270:
1266:
1247:
1243:
1235:
1230:
1226:
1221:
1215:
1211:
1203:
1198:
1194:
1189:
1183:
1179:
1170:
1163:
1159:
1154:
1150:
1145:
1141:
1136:
1134:
1129:
1125:
1121:
1114:
1110:
1103:
1099:
1093:
1089:
1085:such that if
1080:
1076:
1071:
1067:
1062:
1058:
1053:
1027:
1023:
1018:
1012:
1007:Suppose that
1004:
996:
994:
984:
979:
971:
965:
962:
950:
946:
940:
936:
926:
921:
917:
913:
902:
893:
889:
856:
844:
832:
828:
822:
818:
810:
806:
797:
762:
755:
754:
753:
747:
731:
700:
696:
692:
663:
653:
644:
640:
634:
630:
622:
618:
607:
603:
597:
593:
586:
582:
548:
543:
537:
533:
526:
522:
517:
510:
506:
499:
493:
489:
484:
483:
475:
471:
464:
463:
451:
447:
441:
437:
403:
399:
393:
389:
382:
378:
375:
368:
363:
362:
360:
356:
352:
325:
324:
323:
319:
313:
305:
300:
298:
296:
292:
288:
282:
278:
270:
266:
259:
255:
250:
248:
242:
224:
221:
217:
207:
205:
201:
197:
179:
175:
166:
163:
142:
139:
135:
131:
126:
122:
110:
106:
102:
81:
78:
74:
70:
65:
61:
49:
45:
41:
37:
35:
29:
27:
22:
5502:Polynomially
5460:
5431:Grothendieck
5424:tame Fréchet
5374:Bornological
5234:Linear cone
5226:Convex cone
5201:Banach disks
5143:Sesquilinear
4998:Main results
4988:Vector space
4943:Completeness
4938:Banach space
4843:Balanced set
4817:Distribution
4755:Applications
4608:Krein–Milman
4593:Closed graph
4382:
4348:
4314:
4284:
4254:
4227:
4193:
4155:
4128:
4101:
4063:
4025:
3998:
3979:
3947:
3915:
3888:
3854:
3841:20 September
3839:. Retrieved
3834:
3796:
3789:Bibliography
3774:
3762:
3750:
3738:
3726:
3714:
3687:
3675:
3618:
3612:
3566:Direct limit
3341:
3337:
3293:
3289:
3277:, 0, ..., 0)
3273:
3269:
3262:
3257:
3253:
3246:
3242:
3234:
3230:
3225:
3221:
3214:
3210:
3204:
3200:
3195:
3191:
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3158:
3151:
3087:
3085:
3053:
3049:
2887:
2691:
2608:
2606:
2575:
2571:
2475:
2465:
2461:
2443:
2436:
2432:
2405:
2401:
2391:
2379:bornological
2363:bornological
2354:
2348:
2331:bornological
2328:
2312:
2308:
2297:
2293:
2289:
2274:
2267:
2263:
2255:
2251:
2244:
2240:
2232:
2228:
2221:
2217:
2213:direct limit
2212:
2208:
2204:
2148:
2138:
2120:
2109:
2104:
2100:
2095:
2091:
2087:quotient map
2060:
2056:
2049:
2045:
2044:
2033:
2029:
2018:
2014:
2010:
1987:
1976:
1972:
1965:
1960:
1954:
1950:
1945:
1941:
1937:
1933:. That is,
1928:
1924:
1917:
1913:
1909:
1905:
1899:
1893:
1889:
1885:
1878:
1874:
1871:
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1856:
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1747:
1743:
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1734:
1727:
1650:
1646:
1639:
1635:
1628:
1624:
1618:
1614:
1609:
1603:
1565:
1561:
1560:
1556:direct limit
1553:
1539:
1530:
1519:
1514:
1511:
1507:
1503:
1497:
1493:
1491:. The maps
1488:
1485:directed set
1475:
1470:
1465:
1434:
1432:
1321:
1168:
1161:
1157:
1152:
1148:
1143:
1139:
1132:
1127:
1123:
1119:
1112:
1108:
1101:
1097:
1091:
1087:
1084:
1078:
1074:
1069:
1065:
1060:
1056:
1025:
1021:
1017:directed set
1010:
1006:
1003:Direct limit
982:
977:
963:
948:
944:
938:
934:
924:
919:
915:
900:
891:
887:
884:
857:morphism in
842:
830:
826:
820:
816:
808:
804:
760:
729:
698:
694:
660:
658:
651:
642:
638:
632:
628:
620:
616:
605:
601:
595:
591:
584:
541:
535:
531:
524:
520:
518:topology on
515:
508:
504:
491:
487:
481:
480:
473:
469:
449:
445:
439:
435:
401:
397:
391:
387:
380:
374:directed set
321:
294:
290:
286:
280:
276:
268:
264:
257:
253:
246:
244:
240:
208:
203:
199:
195:
109:direct limit
104:
43:
33:
31:
25:
24:
18:
5496:Quasinormed
5409:FK-AK space
5303:Linear span
5298:Convex hull
5283:Affine hull
5086:Almost open
5026:Hahn–Banach
4772:Heat kernel
4762:Hardy space
4669:Trace class
4583:Hahn–Banach
4545:Topological
3767:Trèves 2006
3755:Trèves 2006
3743:Trèves 2006
3731:Trèves 2006
3239:defined by
3199:, consider
2611:-space is,
1775:induced by
1610:Assumptions
198:stands for
21:mathematics
5536:Stereotype
5394:(DF)-space
5389:Convenient
5128:Functional
5096:Continuous
5081:Linear map
5021:F. Riesz's
4963:Linear map
4705:C*-algebra
4520:Properties
4371:1145563701
3245: := (
3086:With this
2426:such that
2357:-space is
2325:Properties
2317:for every
2071:surjective
1903:such that
1508:connecting
855:continuous
802:, the map
752:such that
485:be called
316:See also:
301:Definition
5552:Uniformly
5511:Reflexive
5359:Barrelled
5355:Countably
5267:Symmetric
5165:Transpose
4679:Unbounded
4674:Transpose
4632:Operators
4561:Separable
4556:Reflexive
4541:Algebraic
4527:Barrelled
4401:849801114
4337:316568534
4303:853623322
4283:(2006) .
4246:175294365
4216:840278135
4147:144216834
4120:180577972
4090:840293704
4062:(1983) .
3907:395340485
3853:(1987) .
3819:297140003
3587:Citations
3537:′
3510:′
3492:′
3487:σ
3460:′
3429:′
3359:′
3297:'s into
3166: :=
3108:∞
3016:∞
2908:∞
2783:⋃
2750:⊂
2747:…
2744:⊂
2731:⊂
2728:…
2725:⊂
2712:⊂
2629:∞
2551:′
2519:∞
2383:barrelled
2371:LF spaces
2359:barrelled
2345:LF-spaces
2335:barrelled
2209:countable
2075:bijective
2067:injective
1944: :=
1811: :=
1604:injective
1600:injective
1400:≤
1384:∈
1342:∙
744:, is the
167:and each
5611:Category
5596:Category
5547:Strictly
5521:Schwartz
5461:LF-space
5456:LB-space
5414:FK-space
5384:Complete
5364:BK-space
5289:Relative
5236:(subset)
5228:(subset)
5155:Seminorm
5138:Bilinear
4881:Category
4693:Algebras
4575:Theorems
4532:Complete
4501:Schwartz
4447:glossary
4381:(2013).
4347:(2020).
4319:Elsevier
4273:24909067
4226:(1996).
4100:(1979).
3978:(1966).
3944:(1973).
3934:30593138
3889:Topology
3887:(1966).
3877:17499190
3829:(1988).
3581:LB-space
3561:DF-space
3555:See also
3189:and for
2592:Examples
2586:LB-space
2582:normable
2469:. Every
2099: :
2065:is/is a
1851: :
1733: :
1489:directed
1466:directed
1463:that is
1277:→
1222:→
1190:→
1064: :
812: :
624: :
516:original
361:(TVSs);
355:category
202:imit of
5561:)
5509:)
5451:K-space
5436:Hilbert
5419:Fréchet
5404:F-space
5379:Brauner
5372:)
5357:)
5339:Asplund
5321:)
5291:)
5211:Bounded
5106:Compact
5091:Bounded
5028: (
4684:Unitary
4664:Nuclear
4649:Compact
4644:Bounded
4639:Adjoint
4613:Min–max
4506:Sobolev
4491:Nuclear
4481:Hilbert
4476:Fréchet
4441: (
4082:0248498
4052:8588370
4017:8210342
3576:F-space
3268:, ...,
3252:, ...,
2471:bounded
2452:into a
2333:(resp.
2124:nuclear
2069:(resp.
1996:subset
1872:If the
1622:, each
1512:linking
1504:bonding
1471:indexed
1117:and if
695:colimit
479:should
245:strict
38:, is a
36:)-space
5573:Webbed
5559:Quasi-
5481:Montel
5471:Mackey
5370:Ultra-
5349:Banach
5257:Radial
5221:Convex
5191:Affine
5133:Linear
5101:Closed
4925:(TVSs)
4659:Normal
4496:Orlicz
4486:Hölder
4466:Banach
4455:Spaces
4443:topics
4399:
4389:
4369:
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3807:
3625:
3474:(i.e.
2351:meager
2205:strict
1961:where
430:where
249:-space
42:(TVS)
28:-space
5531:Smith
5516:Riesz
5507:Semi-
5319:Quasi
5313:Polar
4471:Besov
3261:) ↦ (
3194:<
2774:with
1510:, or
1483:is a
1473:) by
1095:then
1015:is a
853:is a
795:, and
107:is a
23:, an
5150:Norm
5074:form
5062:Maps
4819:(or
4537:Dual
4397:OCLC
4387:ISBN
4367:OCLC
4357:ISBN
4333:OCLC
4323:ISBN
4299:OCLC
4289:ISBN
4269:OCLC
4259:ISBN
4242:OCLC
4232:ISBN
4212:OCLC
4202:ISBN
4178:OCLC
4168:ISBN
4143:OCLC
4133:ISBN
4116:OCLC
4106:ISBN
4086:OCLC
4068:ISBN
4048:OCLC
4038:ISBN
4013:OCLC
4003:ISBN
3984:ISBN
3964:OCLC
3954:ISBN
3930:OCLC
3920:ISBN
3903:OCLC
3893:ISBN
3873:OCLC
3863:ISBN
3843:2020
3815:OCLC
3805:ISBN
3623:ISBN
2580:are
2381:and
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590:= (
386:= (
4198:GTM
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2013:∩
1940:+
1912:∈
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1024:≤
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838:→
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604:∈
462:;
400:∈
295:LF
291:LF
287:LF
271:+1
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