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The argument for the existence of discontinuous linear maps on normed spaces can be generalized to all metrizable topological vector spaces, especially to all Fréchet spaces, but there exist infinite-dimensional locally convex topological vector spaces such that every functional is continuous. On the
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of a topological vector space is the collection of continuous linear maps from the space into the underlying field. Thus the failure of some linear maps to be continuous for infinite-dimensional normed spaces implies that for these spaces, one needs to distinguish the algebraic dual space from the
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The upshot is that the existence of discontinuous linear maps depends on AC; it is consistent with set theory without AC that there are no discontinuous linear maps on complete spaces. In particular, no concrete construction such as the derivative can succeed in defining a discontinuous linear map
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So the natural question to ask about linear operators that are not everywhere-defined is whether they are closable. The answer is, "not necessarily"; indeed, every infinite-dimensional normed space admits linear operators that are not closable. As in the case of discontinuous operators considered
473:
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function is continuous (this is to be understood in the terminology of constructivism, according to which only representable functions are considered to be functions). Such stances are held by only a small minority of working mathematicians.
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Notice that by using the fact that any set of linearly independent vectors can be completed to a basis, we implicitly used the axiom of choice, which was not needed for the concrete example in the previous section.
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continuous dual space which is then a proper subset. It illustrates the fact that an extra dose of caution is needed in doing analysis on infinite-dimensional spaces as compared to finite-dimensional ones.
3368:. The upshot is that spaces with fewer convex sets have fewer functionals, and in the worst-case scenario, a space may have no functionals at all other than the zero functional. This is the case for the
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viewpoint. For example, H. G. Garnir, in searching for so-called "dream spaces" (topological vector spaces on which every linear map into a normed space is continuous), was led to adopt ZF +
613:
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Solovay's result shows that it is not necessary to assume that all infinite-dimensional vector spaces admit discontinuous linear maps, and there are schools of analysis which adopt a more
540:
2586:(AC) is used in the general existence theorem of discontinuous linear maps. In fact, there are no constructive examples of discontinuous linear maps with complete domain (for example,
1417:
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This example can be extended into a general theorem about the existence of discontinuous linear maps on any infinite-dimensional normed space (as long as the codomain is not trivial).
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closed operator on a complete domain is continuous, so to obtain a discontinuous closed operator, one must permit operators which are not defined everywhere.
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in which every set of reals is measurable. This implies that there are no discontinuous linear real functions. Clearly AC does not hold in the model.
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which assigns to each function its derivative is similarly discontinuous. Note that although the derivative operator is not continuous, it is
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The fact that the domain is not complete here is important: discontinuous operators on complete spaces require a little more work.
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2590:). In analysis as it is usually practiced by working mathematicians, the axiom of choice is always employed (it is an axiom of
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One can consider even more general spaces. For example, the existence of a homomorphism between complete separable metric
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is infinite-dimensional, to show the existence of a linear functional which is not continuous then amounts to constructing
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Examples of discontinuous linear maps are easy to construct in spaces that are not complete; on any Cauchy sequence
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2597:); thus, to the analyst, all infinite-dimensional topological vector spaces admit discontinuous linear maps.
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468:{\displaystyle \|f(x)\|=\left\|\sum _{i=1}^{n}x_{i}f(e_{i})\right\|\leq \sum _{i=1}^{n}|x_{i}|\|f(e_{i})\|.}
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grow without bound. In a sense, the linear operators are not continuous because the space has "holes".
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above, the proof requires the axiom of choice and so is in general nonconstructive, though again, if
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884:
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That is, in studying operators that are not everywhere-defined, one may restrict one's attention to
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Discontinuous linear maps can be proven to exist more generally, even if the space is complete. Let
17:
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real function is linear if and only if it is measurable, so for every such function there is a
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For example, the weak topology w.r.t. the space of all (algebraically) linear functionals.
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1933:, are linearly independent. One may find a Hamel basis containing them, and define a map
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of linearly independent vectors which does not have a limit, there is a linear operator
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from which it follows that these spaces are nonconvex. Note that here is indicated the
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1905:(note that some authors use this term in a broader sense to mean an algebraic basis of
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3681:(1970), "A model of set-theory in which every set of reals is Lebesgue measurable",
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and with real values, is linear, but not continuous. Indeed, consider the sequence
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is not complete here, as must be the case when there is such a constructible map.
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is infinite-dimensional, this proof will fail as there is no guarantee that the
75:, it is trickier; such maps can be proven to exist, but the proof relies on the
28:
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In fact, there is even an example of a linear operator whose graph has closure
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729:{\displaystyle \|f(x)\|\leq \left(\sum _{i=1}^{n}|x_{i}|\right)M\leq CM\|x\|.}
32:
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2008:
acts as the identity on the rest of the Hamel basis, and extend to all of
1686:{\displaystyle T(f_{n})={\frac {n^{2}\cos(n^{2}\cdot 0)}{n}}=n\to \infty }
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1595:. This sequence converges uniformly to the constantly zero function, but
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is the zero map which is trivially continuous. In all other cases, when
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so this provides a sort of maximally discontinuous linear map (confer
3696:
43:
and are often used as approximations to more general functions (see
3651: – A vector space with a topology defined by convex open sets
83:
A linear map from a finite-dimensional space is always continuous
4270:
3786:
2292:, which will imply the existence of a discontinuous linear map
2280:
is not the zero space. We will find a discontinuous linear map
2635:
which states, among other things, that any linear map from an
2591:
2570:, and since it is clearly not bounded, it is not continuous.
1180:
is not the zero space, one can find a discontinuous map from
760:
and so is continuous. In fact, to see this, simply note that
3625:{\displaystyle \|f\|=\int _{I}{\frac {|f(x)|}{1+|f(x)|}}dx.}
2664:
Many naturally occurring linear discontinuous operators are
2550:
Complete this sequence of linearly independent vectors to a
3277:() respectively, and so normed spaces. Define an operator
2820:{\displaystyle T:\operatorname {Dom} (T)\subseteq X\to Y.}
1011:{\displaystyle f(B(x,\delta ))\subseteq B(f(x),\epsilon )}
621:
any two norms on a finite-dimensional space are equivalent
55:), then it makes sense to ask whether all linear maps are
3293:) on to the same function on . As a consequence of the
3632:
This non-locally convex space has a trivial dual space.
870:{\displaystyle \|f(x)-f(x')\|=\|f(x-x')\|\leq K\|x-x'\|}
3653:
Pages displaying short descriptions of redirect targets
2631:
is a negation of strong AC) as his axioms to prove the
205:{\displaystyle \left(e_{1},e_{2},\ldots ,e_{n}\right)}
63:
topological vector spaces (e.g., infinite-dimensional
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3177:is not complete, there are constructible examples.
2566:so defined will extend uniquely to a linear map on
4142:Spectral theory of ordinary differential equations
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1562:{\displaystyle f_{n}(x)={\frac {\sin(n^{2}x)}{n}}}
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294:{\displaystyle f(x)=\sum _{i=1}^{n}x_{i}f(e_{i}),}
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3518:Another such example is the space of real-valued
2639:to a TVS is continuous. Going to the extreme of
1778:, as would hold for a continuous map. Note that
608:{\displaystyle \sum _{i=1}^{n}|x_{i}|\leq C\|x\|}
59:. It turns out that for maps defined on infinite-
2045:be any sequence of rationals which converges to
1357:
489:
2562:at the other vectors in the basis to be zero.
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2627:(dependent choice is a weakened form and the
216:which may be taken to be unit vectors. Then,
8:
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1164:is the zero space {0}, the only map between
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535:{\displaystyle M=\sup _{i}\{\|f(e_{i})\|\},}
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3247:the space of polynomial functions from to
2400:which is not bounded. For that, consider a
67:), the answer is generally no: there exist
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3791:
3783:
3660: – Type of function in linear algebra
1412:{\displaystyle \|f\|=\sup _{x\in }|f(x)|.}
79:and does not provide an explicit example.
39:which preserve the algebraic structure of
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4095:Group algebra of a locally compact group
3769:Handbook of Analysis and its Foundations
3721:Handbook of Analysis and Its Foundations
3056:{\displaystyle {\overline {\Gamma (T)}}}
2989:{\displaystyle {\overline {\Gamma (T)}}}
2859:{\displaystyle \operatorname {Dom} (T).}
2767:{\displaystyle \operatorname {Dom} (T),}
3670:
3213:Such an operator is not closable. Let
937:{\displaystyle \delta \leq \epsilon /K}
619:>0 which follows from the fact that
4428:Uniform boundedness (Banach–Steinhaus)
3515:which do have nontrivial dual spaces.
3409:{\displaystyle L^{p}(\mathbb {R} ,dx)}
2453:, which we normalize. Then, we define
1299:{\displaystyle \|T(e_{i})\|/\|e_{i}\|}
3639:can also be shown nonconstructively.
3063:is itself the graph of some operator
2505:{\displaystyle T(e_{n})=n\|e_{n}\|\,}
1798:is real-valued, and so is actually a
7:
3281:which takes the polynomial function
47:). If the spaces involved are also
35:form an important class of "simple"
18:A linear map which is not continuous
3456:on the real line. There are other
1960:{\displaystyle f:\mathbb {R} \to R}
3752:Constantin Costara, Dumitru Popa,
3035:
2968:
2958:. Otherwise, consider its closure
2880:
2633:Garnir–Wright closed graph theorem
2385:is an arbitrary nonzero vector in
2166:), but not continuous. Note that
1909:vector space). Note that any two
1771:{\displaystyle T(f_{n})\to T(0)=0}
1708:
1680:
71:. If the domain of definition is
25:
2827:We don't lose much if we replace
1090:{\displaystyle B(f(x),\epsilon )}
4966:
4965:
4251:
4250:
4177:Topological quantum field theory
3754:Exercises in Functional Analysis
3086:{\displaystyle {\overline {T}},}
2656:everywhere on a complete space.
1309:For example, consider the space
4953:With the approximation property
3724:, Academic Press, p. 136,
3136:{\displaystyle {\overline {T}}}
2265:{\displaystyle K=\mathbb {C} .}
2186:relies on the axiom of choice.
903:{\displaystyle \epsilon >0,}
4416:Open mapping (Banach–Schauder)
3649:Finest locally convex topology
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3973:Uniform boundedness principle
2543:{\displaystyle n=1,2,\ldots }
1822:(an element of the algebraic
3445:{\displaystyle 0<p<1,}
3262:{\displaystyle \mathbb {R} }
3236:{\displaystyle \mathbb {R} }
3128:
3075:
3048:
2981:
2870:without loss of generality.
2276:is infinite-dimensional and
2159:{\displaystyle \mathbb {R} }
2137:{\displaystyle \mathbb {Q} }
2023:{\displaystyle \mathbb {R} }
1714:{\displaystyle n\to \infty }
1469:{\displaystyle T(f)=f'(0)\,}
1176:is infinite-dimensional and
1097:are the normed balls around
1046:{\displaystyle B(x,\delta )}
877:for some universal constant
4637:Radially convex/Star-shaped
4622:Pre-compact/Totally bounded
3508:{\displaystyle 0<p<1}
3328:nowhere continuous function
2600:On the other hand, in 1970
2578:Role of the axiom of choice
1897:as a vector space over the
1893:An algebraic basis for the
1146:), which gives continuity.
5018:
4323:Continuous linear operator
4116:Invariant subspace problem
3522:on the unit interval with
3319:{\displaystyle X\times Y,}
3206:{\displaystyle X\times Y.}
3018:{\displaystyle X\times Y.}
2944:{\displaystyle X\times Y,}
2895:{\displaystyle \Gamma (T)}
2111:{\displaystyle f(\pi )=0.}
1998:{\displaystyle f(\pi )=0,}
1333:on the interval with the
4961:
4668:Algebraic interior (core)
4410:Vector-valued Hahn–Banach
4298:Topological vector spaces
4246:
3836:
3295:Stone–Weierstrass theorem
3269:. They are subspaces of
2868:densely defined operators
2679:To be more concrete, let
2193:General existence theorem
1889:A nonconstructive example
1243:such that the quantities
764:is linear, and therefore
95:be two normed spaces and
69:discontinuous linear maps
53:topological vector spaces
4498:Topological homomorphism
4358:Topological vector space
4085:Spectrum of a C*-algebra
3771:, Academic Press, 1997.
3718:Schechter, Eric (1996),
3364:associates a continuous
542:and using the fact that
120:{\displaystyle f:X\to Y}
4182:Noncommutative geometry
2438:{\displaystyle n\geq 1}
2182:. The construction of
1588:{\displaystyle n\geq 1}
758:bounded linear operator
5002:Functions and mappings
4556:Absolutely convex/disk
4238:Tomita–Takesaki theory
4213:Approximation property
4157:Calculus of variations
3626:
3509:
3477:
3446:
3410:
3338:Impact for dual spaces
3320:
3263:
3237:
3207:
3164:
3137:
3106:
3087:
3057:
3019:
2990:
2945:
2916:
2896:
2860:
2821:
2768:
2733:
2713:
2693:
2544:
2506:
2439:
2379:
2352:
2266:
2235:
2160:
2138:
2112:
2059:
2024:
1999:
1961:
1927:
1872:
1871:{\displaystyle X\to X}
1846:
1816:
1792:
1772:
1715:
1687:
1589:
1563:
1490:
1470:
1413:
1323:
1300:
1237:
1217:
1140:
1111:
1091:
1047:
1012:
938:
904:
871:
750:
730:
676:
609:
569:
536:
469:
416:
358:
295:
258:
206:
121:
4591:Complemented subspace
4405:hyperplane separation
4233:Banach–Mazur distance
4196:Generalized functions
3684:Annals of Mathematics
3627:
3510:
3478:
3476:{\displaystyle L^{p}}
3447:
3411:
3321:
3264:
3238:
3208:
3165:
3138:
3107:
3088:
3058:
3020:
2991:
2946:
2917:
2897:
2861:
2822:
2769:
2734:
2714:
2694:
2545:
2507:
2440:
2380:
2378:{\displaystyle y_{0}}
2353:
2304:given by the formula
2267:
2236:
2161:
2139:
2113:
2060:
2025:
2000:
1962:
1928:
1873:
1847:
1845:{\displaystyle X^{*}}
1817:
1793:
1773:
1716:
1688:
1590:
1564:
1491:
1471:
1414:
1324:
1301:
1238:
1218:
1216:{\displaystyle e_{i}}
1141:
1112:
1092:
1048:
1013:
939:
905:
872:
751:
731:
656:
610:
549:
537:
470:
396:
338:
296:
238:
207:
122:
4841:Locally convex space
4391:Closed graph theorem
4343:Locally convex space
3978:Kakutani fixed-point
3963:Riesz representation
3530:
3520:measurable functions
3487:
3460:
3421:
3373:
3351:Beyond normed spaces
3301:
3251:
3225:
3219:polynomial functions
3188:
3151:
3120:
3096:
3067:
3029:
3000:
2962:
2926:
2906:
2877:
2835:
2778:
2743:
2723:
2703:
2683:
2670:closed graph theorem
2647:, which states that
2582:As noted above, the
2516:
2457:
2447:linearly independent
2423:
2362:
2308:
2245:
2217:
2148:
2126:
2087:
2058:{\displaystyle \pi }
2049:
2030:by linearity. Let {
2012:
1971:
1937:
1926:{\displaystyle \pi }
1917:
1856:
1829:
1806:
1782:
1725:
1699:
1599:
1573:
1500:
1480:
1430:
1341:
1313:
1247:
1227:
1200:
1139:{\displaystyle f(x)}
1121:
1101:
1057:
1022:
948:
914:
885:
768:
740:
627:
546:
479:
309:
220:
147:
99:
45:linear approximation
4992:Functional analysis
4821:Interpolation space
4353:Operator topologies
4162:Functional calculus
4121:Mahler's conjecture
4100:Von Neumann algebra
3814:Functional analysis
3358:Hahn–Banach theorem
1913:numbers, say 1 and
1852:). The linear map
303:triangle inequality
4851:(Pseudo)Metrizable
4683:Minkowski addition
4535:Sublinear function
4187:Riemann hypothesis
3886:Topological vector
3756:, Springer, 2003.
3679:Solovay, Robert M.
3658:Sublinear function
3622:
3505:
3473:
3442:
3406:
3316:
3259:
3233:
3203:
3163:{\displaystyle T.}
3160:
3133:
3102:
3083:
3053:
3015:
2986:
2941:
2912:
2892:
2856:
2831:by the closure of
2817:
2764:
2729:
2709:
2689:
2674:everywhere-defined
2552:vector space basis
2540:
2502:
2435:
2375:
2348:
2262:
2231:
2156:
2134:
2108:
2055:
2020:
1995:
1957:
1923:
1868:
1842:
1812:
1788:
1768:
1711:
1683:
1585:
1559:
1486:
1466:
1409:
1383:
1319:
1296:
1233:
1213:
1192:A concrete example
1136:
1107:
1087:
1043:
1008:
934:
900:
867:
746:
726:
605:
532:
497:
465:
291:
202:
141:finite-dimensional
127:a linear map from
117:
49:topological spaces
4979:
4978:
4698:Relative interior
4444:Bilinear operator
4328:Linear functional
4264:
4263:
4167:Integral operator
3944:
3943:
3767:Schechter, Eric,
3687:, Second Series,
3611:
3366:linear functional
3131:
3105:{\displaystyle T}
3078:
3051:
2984:
2915:{\displaystyle T}
2732:{\displaystyle Y}
2712:{\displaystyle X}
2692:{\displaystyle T}
2602:Robert M. Solovay
2118:By construction,
1815:{\displaystyle X}
1800:linear functional
1791:{\displaystyle T}
1669:
1557:
1489:{\displaystyle X}
1356:
1322:{\displaystyle X}
1236:{\displaystyle T}
1110:{\displaystyle x}
749:{\displaystyle f}
488:
143:, choose a basis
16:(Redirected from
5009:
4969:
4968:
4943:Uniformly smooth
4612:
4604:
4571:Balanced/Circled
4561:Absorbing/Radial
4291:
4284:
4277:
4268:
4254:
4253:
4172:Jones polynomial
4090:Operator algebra
3834:
3807:
3800:
3793:
3784:
3745:
3742:
3736:
3734:
3715:
3709:
3707:
3675:
3654:
3631:
3629:
3628:
3623:
3612:
3610:
3609:
3592:
3580:
3579:
3562:
3556:
3554:
3553:
3514:
3512:
3511:
3506:
3482:
3480:
3479:
3474:
3472:
3471:
3454:Lebesgue measure
3451:
3449:
3448:
3443:
3415:
3413:
3412:
3407:
3393:
3385:
3384:
3356:other hand, the
3325:
3323:
3322:
3317:
3268:
3266:
3265:
3260:
3258:
3242:
3240:
3239:
3234:
3232:
3217:be the space of
3212:
3210:
3209:
3204:
3169:
3167:
3166:
3161:
3142:
3140:
3139:
3134:
3132:
3124:
3111:
3109:
3108:
3103:
3092:
3090:
3089:
3084:
3079:
3071:
3062:
3060:
3059:
3054:
3052:
3047:
3033:
3024:
3022:
3021:
3016:
2995:
2993:
2992:
2987:
2985:
2980:
2966:
2950:
2948:
2947:
2942:
2921:
2919:
2918:
2913:
2901:
2899:
2898:
2893:
2865:
2863:
2862:
2857:
2826:
2824:
2823:
2818:
2773:
2771:
2770:
2765:
2738:
2736:
2735:
2730:
2718:
2716:
2715:
2710:
2698:
2696:
2695:
2690:
2672:asserts that an
2660:Closed operators
2645:Ceitin's theorem
2549:
2547:
2546:
2541:
2511:
2509:
2508:
2503:
2497:
2496:
2475:
2474:
2444:
2442:
2441:
2436:
2384:
2382:
2381:
2376:
2374:
2373:
2357:
2355:
2354:
2349:
2347:
2346:
2271:
2269:
2268:
2263:
2258:
2240:
2238:
2237:
2232:
2230:
2165:
2163:
2162:
2157:
2155:
2143:
2141:
2140:
2135:
2133:
2117:
2115:
2114:
2109:
2064:
2062:
2061:
2056:
2029:
2027:
2026:
2021:
2019:
2004:
2002:
2001:
1996:
1966:
1964:
1963:
1958:
1950:
1932:
1930:
1929:
1924:
1911:noncommensurable
1877:
1875:
1874:
1869:
1851:
1849:
1848:
1843:
1841:
1840:
1821:
1819:
1818:
1813:
1797:
1795:
1794:
1789:
1777:
1775:
1774:
1769:
1743:
1742:
1720:
1718:
1717:
1712:
1692:
1690:
1689:
1684:
1670:
1665:
1655:
1654:
1636:
1635:
1625:
1617:
1616:
1594:
1592:
1591:
1586:
1568:
1566:
1565:
1560:
1558:
1553:
1546:
1545:
1526:
1512:
1511:
1495:
1493:
1492:
1487:
1475:
1473:
1472:
1467:
1455:
1418:
1416:
1415:
1410:
1405:
1388:
1382:
1331:smooth functions
1328:
1326:
1325:
1320:
1305:
1303:
1302:
1297:
1292:
1291:
1279:
1268:
1267:
1242:
1240:
1239:
1234:
1222:
1220:
1219:
1214:
1212:
1211:
1145:
1143:
1142:
1137:
1116:
1114:
1113:
1108:
1096:
1094:
1093:
1088:
1052:
1050:
1049:
1044:
1017:
1015:
1014:
1009:
943:
941:
940:
935:
930:
909:
907:
906:
901:
876:
874:
873:
868:
863:
834:
802:
755:
753:
752:
747:
735:
733:
732:
727:
701:
697:
696:
691:
690:
681:
675:
670:
614:
612:
611:
606:
589:
584:
583:
574:
568:
563:
541:
539:
538:
533:
519:
518:
496:
474:
472:
471:
466:
455:
454:
436:
431:
430:
421:
415:
410:
392:
388:
384:
383:
368:
367:
357:
352:
300:
298:
297:
292:
284:
283:
268:
267:
257:
252:
211:
209:
208:
203:
201:
197:
196:
195:
177:
176:
164:
163:
126:
124:
123:
118:
21:
5017:
5016:
5012:
5011:
5010:
5008:
5007:
5006:
4997:Axiom of choice
4982:
4981:
4980:
4975:
4957:
4719:B-complete/Ptak
4702:
4646:
4610:
4602:
4581:Bounding points
4544:
4486:Densely defined
4432:
4421:Bounded inverse
4367:
4301:
4295:
4265:
4260:
4242:
4206:Advanced topics
4201:
4125:
4104:
4063:
4029:Hilbert–Schmidt
4002:
3993:Gelfand–Naimark
3940:
3890:
3825:
3811:
3749:
3748:
3743:
3739:
3732:
3717:
3716:
3712:
3697:10.2307/1970696
3677:
3676:
3672:
3667:
3652:
3645:
3581:
3557:
3545:
3528:
3527:
3485:
3484:
3463:
3458:
3457:
3419:
3418:
3376:
3371:
3370:
3362:Minkowski gauge
3353:
3340:
3299:
3298:
3249:
3248:
3223:
3222:
3186:
3185:
3149:
3148:
3118:
3117:
3094:
3093:
3065:
3064:
3034:
3027:
3026:
2998:
2997:
2967:
2960:
2959:
2924:
2923:
2904:
2903:
2875:
2874:
2833:
2832:
2776:
2775:
2741:
2740:
2721:
2720:
2701:
2700:
2681:
2680:
2662:
2584:axiom of choice
2580:
2514:
2513:
2488:
2466:
2455:
2454:
2421:
2420:
2418:
2412:
2365:
2360:
2359:
2338:
2306:
2305:
2243:
2242:
2215:
2214:
2209:over the field
2195:
2146:
2145:
2124:
2123:
2122:is linear over
2085:
2084:
2082:
2070:
2047:
2046:
2044:
2038:
2010:
2009:
1969:
1968:
1935:
1934:
1915:
1914:
1891:
1854:
1853:
1832:
1827:
1826:
1804:
1803:
1780:
1779:
1734:
1723:
1722:
1697:
1696:
1646:
1627:
1626:
1608:
1597:
1596:
1571:
1570:
1537:
1527:
1503:
1498:
1497:
1478:
1477:
1448:
1428:
1427:
1339:
1338:
1329:of real-valued
1311:
1310:
1283:
1259:
1245:
1244:
1225:
1224:
1203:
1198:
1197:
1194:
1119:
1118:
1099:
1098:
1055:
1054:
1020:
1019:
946:
945:
912:
911:
883:
882:
881:. Thus for any
856:
827:
795:
766:
765:
738:
737:
682:
655:
651:
625:
624:
575:
544:
543:
510:
477:
476:
446:
422:
375:
359:
337:
333:
307:
306:
275:
259:
218:
217:
187:
168:
155:
154:
150:
145:
144:
97:
96:
85:
77:axiom of choice
23:
22:
15:
12:
11:
5:
5015:
5013:
5005:
5004:
4999:
4994:
4984:
4983:
4977:
4976:
4974:
4973:
4962:
4959:
4958:
4956:
4955:
4950:
4945:
4940:
4938:Ultrabarrelled
4930:
4924:
4919:
4913:
4908:
4903:
4898:
4893:
4888:
4879:
4873:
4868:
4866:Quasi-complete
4863:
4861:Quasibarrelled
4858:
4853:
4848:
4843:
4838:
4833:
4828:
4823:
4818:
4813:
4808:
4803:
4802:
4801:
4791:
4786:
4781:
4776:
4771:
4766:
4761:
4756:
4751:
4741:
4736:
4726:
4721:
4716:
4710:
4708:
4704:
4703:
4701:
4700:
4690:
4685:
4680:
4675:
4670:
4660:
4654:
4652:
4651:Set operations
4648:
4647:
4645:
4644:
4639:
4634:
4629:
4624:
4619:
4614:
4606:
4598:
4593:
4588:
4583:
4578:
4573:
4568:
4563:
4558:
4552:
4550:
4546:
4545:
4543:
4542:
4537:
4532:
4527:
4522:
4521:
4520:
4515:
4510:
4500:
4495:
4494:
4493:
4488:
4483:
4478:
4473:
4468:
4463:
4453:
4452:
4451:
4440:
4438:
4434:
4433:
4431:
4430:
4425:
4424:
4423:
4413:
4407:
4398:
4393:
4388:
4386:Banach–Alaoglu
4383:
4381:Anderson–Kadec
4377:
4375:
4369:
4368:
4366:
4365:
4360:
4355:
4350:
4345:
4340:
4335:
4330:
4325:
4320:
4315:
4309:
4307:
4306:Basic concepts
4303:
4302:
4296:
4294:
4293:
4286:
4279:
4271:
4262:
4261:
4259:
4258:
4247:
4244:
4243:
4241:
4240:
4235:
4230:
4225:
4223:Choquet theory
4220:
4215:
4209:
4207:
4203:
4202:
4200:
4199:
4189:
4184:
4179:
4174:
4169:
4164:
4159:
4154:
4149:
4144:
4139:
4133:
4131:
4127:
4126:
4124:
4123:
4118:
4112:
4110:
4106:
4105:
4103:
4102:
4097:
4092:
4087:
4082:
4077:
4075:Banach algebra
4071:
4069:
4065:
4064:
4062:
4061:
4056:
4051:
4046:
4041:
4036:
4031:
4026:
4021:
4016:
4010:
4008:
4004:
4003:
4001:
4000:
3998:Banach–Alaoglu
3995:
3990:
3985:
3980:
3975:
3970:
3965:
3960:
3954:
3952:
3946:
3945:
3942:
3941:
3939:
3938:
3933:
3928:
3926:Locally convex
3923:
3909:
3904:
3898:
3896:
3892:
3891:
3889:
3888:
3883:
3878:
3873:
3868:
3863:
3858:
3853:
3848:
3843:
3837:
3831:
3827:
3826:
3812:
3810:
3809:
3802:
3795:
3787:
3781:
3780:
3765:
3747:
3746:
3737:
3730:
3710:
3669:
3668:
3666:
3663:
3662:
3661:
3655:
3644:
3641:
3621:
3618:
3615:
3608:
3604:
3601:
3598:
3595:
3591:
3587:
3584:
3578:
3574:
3571:
3568:
3565:
3561:
3552:
3548:
3544:
3541:
3538:
3535:
3504:
3501:
3498:
3495:
3492:
3470:
3466:
3441:
3438:
3435:
3432:
3429:
3426:
3405:
3402:
3399:
3396:
3392:
3388:
3383:
3379:
3352:
3349:
3339:
3336:
3330:). Note that
3315:
3312:
3309:
3306:
3257:
3231:
3202:
3199:
3196:
3193:
3159:
3156:
3143:is called the
3130:
3127:
3101:
3082:
3077:
3074:
3050:
3046:
3043:
3040:
3037:
3014:
3011:
3008:
3005:
2983:
2979:
2976:
2973:
2970:
2940:
2937:
2934:
2931:
2911:
2891:
2888:
2885:
2882:
2855:
2852:
2849:
2846:
2843:
2840:
2816:
2813:
2810:
2807:
2804:
2801:
2798:
2795:
2792:
2789:
2786:
2783:
2763:
2760:
2757:
2754:
2751:
2748:
2728:
2708:
2699:be a map from
2688:
2661:
2658:
2641:constructivism
2629:Baire property
2617:constructivist
2579:
2576:
2539:
2536:
2533:
2530:
2527:
2524:
2521:
2500:
2495:
2491:
2487:
2484:
2481:
2478:
2473:
2469:
2465:
2462:
2434:
2431:
2428:
2414:
2408:
2372:
2368:
2345:
2341:
2337:
2334:
2331:
2328:
2325:
2322:
2319:
2316:
2313:
2261:
2257:
2253:
2250:
2229:
2225:
2222:
2194:
2191:
2154:
2132:
2107:
2104:
2101:
2098:
2095:
2092:
2078:
2066:
2054:
2040:
2034:
2018:
1994:
1991:
1988:
1985:
1982:
1979:
1976:
1956:
1953:
1949:
1945:
1942:
1922:
1901:is known as a
1890:
1887:
1867:
1864:
1861:
1839:
1835:
1811:
1787:
1767:
1764:
1761:
1758:
1755:
1752:
1749:
1746:
1741:
1737:
1733:
1730:
1710:
1707:
1704:
1682:
1679:
1676:
1673:
1668:
1664:
1661:
1658:
1653:
1649:
1645:
1642:
1639:
1634:
1630:
1623:
1620:
1615:
1611:
1607:
1604:
1584:
1581:
1578:
1556:
1552:
1549:
1544:
1540:
1536:
1533:
1530:
1524:
1521:
1518:
1515:
1510:
1506:
1485:
1464:
1461:
1458:
1454:
1451:
1447:
1444:
1441:
1438:
1435:
1426:map, given by
1408:
1404:
1400:
1397:
1394:
1391:
1387:
1381:
1378:
1375:
1372:
1369:
1366:
1363:
1359:
1355:
1352:
1349:
1346:
1318:
1295:
1290:
1286:
1282:
1278:
1274:
1271:
1266:
1262:
1258:
1255:
1252:
1232:
1210:
1206:
1193:
1190:
1135:
1132:
1129:
1126:
1106:
1086:
1083:
1080:
1077:
1074:
1071:
1068:
1065:
1062:
1042:
1039:
1036:
1033:
1030:
1027:
1007:
1004:
1001:
998:
995:
992:
989:
986:
983:
980:
977:
974:
971:
968:
965:
962:
959:
956:
953:
933:
929:
925:
922:
919:
910:we can choose
899:
896:
893:
890:
866:
862:
859:
855:
852:
849:
846:
843:
840:
837:
833:
830:
826:
823:
820:
817:
814:
811:
808:
805:
801:
798:
794:
791:
788:
785:
782:
779:
776:
773:
745:
725:
722:
719:
716:
713:
710:
707:
704:
700:
695:
689:
685:
680:
674:
669:
666:
663:
659:
654:
650:
647:
644:
641:
638:
635:
632:
604:
601:
598:
595:
592:
588:
582:
578:
573:
567:
562:
559:
556:
552:
531:
528:
525:
522:
517:
513:
509:
506:
503:
500:
495:
491:
487:
484:
464:
461:
458:
453:
449:
445:
442:
439:
435:
429:
425:
420:
414:
409:
406:
403:
399:
395:
391:
387:
382:
378:
374:
371:
366:
362:
356:
351:
348:
345:
341:
336:
332:
329:
326:
323:
320:
317:
314:
301:and so by the
290:
287:
282:
278:
274:
271:
266:
262:
256:
251:
248:
245:
241:
237:
234:
231:
228:
225:
200:
194:
190:
186:
183:
180:
175:
171:
167:
162:
158:
153:
116:
113:
110:
107:
104:
84:
81:
24:
14:
13:
10:
9:
6:
4:
3:
2:
5014:
5003:
5000:
4998:
4995:
4993:
4990:
4989:
4987:
4972:
4964:
4963:
4960:
4954:
4951:
4949:
4946:
4944:
4941:
4939:
4935:
4931:
4929:) convex
4928:
4925:
4923:
4920:
4918:
4914:
4912:
4909:
4907:
4904:
4902:
4901:Semi-complete
4899:
4897:
4894:
4892:
4889:
4887:
4883:
4880:
4878:
4874:
4872:
4869:
4867:
4864:
4862:
4859:
4857:
4854:
4852:
4849:
4847:
4844:
4842:
4839:
4837:
4834:
4832:
4829:
4827:
4824:
4822:
4819:
4817:
4816:Infrabarreled
4814:
4812:
4809:
4807:
4804:
4800:
4797:
4796:
4795:
4792:
4790:
4787:
4785:
4782:
4780:
4777:
4775:
4774:Distinguished
4772:
4770:
4767:
4765:
4762:
4760:
4757:
4755:
4752:
4750:
4746:
4742:
4740:
4737:
4735:
4731:
4727:
4725:
4722:
4720:
4717:
4715:
4712:
4711:
4709:
4707:Types of TVSs
4705:
4699:
4695:
4691:
4689:
4686:
4684:
4681:
4679:
4676:
4674:
4671:
4669:
4665:
4661:
4659:
4656:
4655:
4653:
4649:
4643:
4640:
4638:
4635:
4633:
4630:
4628:
4627:Prevalent/Shy
4625:
4623:
4620:
4618:
4617:Extreme point
4615:
4613:
4607:
4605:
4599:
4597:
4594:
4592:
4589:
4587:
4584:
4582:
4579:
4577:
4574:
4572:
4569:
4567:
4564:
4562:
4559:
4557:
4554:
4553:
4551:
4549:Types of sets
4547:
4541:
4538:
4536:
4533:
4531:
4528:
4526:
4523:
4519:
4516:
4514:
4511:
4509:
4506:
4505:
4504:
4501:
4499:
4496:
4492:
4491:Discontinuous
4489:
4487:
4484:
4482:
4479:
4477:
4474:
4472:
4469:
4467:
4464:
4462:
4459:
4458:
4457:
4454:
4450:
4447:
4446:
4445:
4442:
4441:
4439:
4435:
4429:
4426:
4422:
4419:
4418:
4417:
4414:
4411:
4408:
4406:
4402:
4399:
4397:
4394:
4392:
4389:
4387:
4384:
4382:
4379:
4378:
4376:
4374:
4370:
4364:
4361:
4359:
4356:
4354:
4351:
4349:
4348:Metrizability
4346:
4344:
4341:
4339:
4336:
4334:
4333:Fréchet space
4331:
4329:
4326:
4324:
4321:
4319:
4316:
4314:
4311:
4310:
4308:
4304:
4299:
4292:
4287:
4285:
4280:
4278:
4273:
4272:
4269:
4257:
4249:
4248:
4245:
4239:
4236:
4234:
4231:
4229:
4228:Weak topology
4226:
4224:
4221:
4219:
4216:
4214:
4211:
4210:
4208:
4204:
4197:
4193:
4190:
4188:
4185:
4183:
4180:
4178:
4175:
4173:
4170:
4168:
4165:
4163:
4160:
4158:
4155:
4153:
4152:Index theorem
4150:
4148:
4145:
4143:
4140:
4138:
4135:
4134:
4132:
4128:
4122:
4119:
4117:
4114:
4113:
4111:
4109:Open problems
4107:
4101:
4098:
4096:
4093:
4091:
4088:
4086:
4083:
4081:
4078:
4076:
4073:
4072:
4070:
4066:
4060:
4057:
4055:
4052:
4050:
4047:
4045:
4042:
4040:
4037:
4035:
4032:
4030:
4027:
4025:
4022:
4020:
4017:
4015:
4012:
4011:
4009:
4005:
3999:
3996:
3994:
3991:
3989:
3986:
3984:
3981:
3979:
3976:
3974:
3971:
3969:
3966:
3964:
3961:
3959:
3956:
3955:
3953:
3951:
3947:
3937:
3934:
3932:
3929:
3927:
3924:
3921:
3917:
3913:
3910:
3908:
3905:
3903:
3900:
3899:
3897:
3893:
3887:
3884:
3882:
3879:
3877:
3874:
3872:
3869:
3867:
3864:
3862:
3859:
3857:
3854:
3852:
3849:
3847:
3844:
3842:
3839:
3838:
3835:
3832:
3828:
3823:
3819:
3815:
3808:
3803:
3801:
3796:
3794:
3789:
3788:
3785:
3778:
3777:0-12-622760-8
3774:
3770:
3766:
3763:
3762:1-4020-1560-7
3759:
3755:
3751:
3750:
3741:
3738:
3733:
3731:9780080532998
3727:
3723:
3722:
3714:
3711:
3706:
3702:
3698:
3694:
3690:
3686:
3685:
3680:
3674:
3671:
3664:
3659:
3656:
3650:
3647:
3646:
3642:
3640:
3638:
3633:
3619:
3616:
3613:
3599:
3593:
3585:
3582:
3569:
3563:
3550:
3546:
3542:
3536:
3525:
3521:
3516:
3502:
3499:
3496:
3493:
3490:
3468:
3464:
3455:
3439:
3436:
3433:
3430:
3427:
3424:
3416:
3400:
3397:
3394:
3381:
3377:
3367:
3363:
3359:
3350:
3348:
3345:
3337:
3335:
3333:
3329:
3313:
3310:
3307:
3304:
3296:
3292:
3288:
3284:
3280:
3276:
3272:
3246:
3220:
3216:
3200:
3197:
3194:
3191:
3183:
3178:
3176:
3170:
3157:
3154:
3146:
3125:
3115:
3099:
3080:
3072:
3041:
3012:
3009:
3006:
3003:
2974:
2957:
2954:
2938:
2935:
2932:
2929:
2922:is closed in
2909:
2886:
2873:If the graph
2871:
2869:
2853:
2847:
2841:
2838:
2830:
2814:
2811:
2805:
2802:
2796:
2790:
2787:
2784:
2781:
2761:
2755:
2749:
2746:
2726:
2706:
2686:
2677:
2675:
2671:
2667:
2659:
2657:
2653:
2650:
2646:
2642:
2638:
2634:
2630:
2626:
2622:
2618:
2613:
2611:
2607:
2603:
2598:
2596:
2593:
2589:
2588:Banach spaces
2585:
2577:
2575:
2571:
2569:
2565:
2561:
2557:
2553:
2537:
2534:
2531:
2528:
2525:
2522:
2519:
2493:
2489:
2482:
2479:
2471:
2467:
2460:
2452:
2448:
2432:
2429:
2426:
2417:
2411:
2407:
2403:
2399:
2395:
2390:
2388:
2370:
2366:
2343:
2339:
2332:
2326:
2323:
2317:
2311:
2303:
2299:
2295:
2291:
2287:
2283:
2279:
2275:
2259:
2251:
2248:
2223:
2220:
2212:
2208:
2207:normed spaces
2204:
2200:
2192:
2190:
2187:
2185:
2181:
2177:
2173:
2169:
2121:
2105:
2102:
2096:
2090:
2081:
2077:
2073:
2069:
2052:
2043:
2037:
2033:
2007:
1992:
1989:
1986:
1980:
1974:
1954:
1943:
1940:
1920:
1912:
1908:
1904:
1900:
1896:
1888:
1886:
1883:
1881:
1865:
1859:
1837:
1833:
1825:
1809:
1801:
1785:
1765:
1762:
1756:
1750:
1739:
1735:
1728:
1702:
1693:
1674:
1671:
1666:
1659:
1656:
1651:
1647:
1640:
1637:
1632:
1628:
1621:
1613:
1609:
1602:
1582:
1579:
1576:
1554:
1547:
1542:
1538:
1531:
1528:
1522:
1516:
1508:
1504:
1483:
1459:
1452:
1449:
1445:
1439:
1433:
1425:
1423:
1406:
1395:
1389:
1376:
1373:
1370:
1364:
1361:
1353:
1347:
1336:
1332:
1316:
1307:
1288:
1284:
1276:
1264:
1260:
1253:
1230:
1208:
1204:
1191:
1189:
1187:
1183:
1179:
1175:
1171:
1167:
1163:
1159:
1156:
1152:
1147:
1130:
1124:
1104:
1081:
1078:
1072:
1066:
1060:
1037:
1034:
1031:
1025:
1002:
999:
993:
987:
981:
978:
969:
966:
963:
957:
951:
931:
927:
923:
920:
917:
897:
894:
891:
888:
880:
860:
857:
853:
850:
844:
841:
831:
828:
824:
821:
815:
809:
799:
796:
789:
786:
780:
774:
763:
759:
743:
723:
717:
711:
708:
705:
702:
698:
687:
683:
672:
667:
664:
661:
657:
652:
648:
639:
633:
623:, one finds
622:
618:
599:
593:
590:
580:
576:
565:
560:
557:
554:
550:
529:
515:
511:
504:
493:
485:
482:
462:
451:
447:
440:
427:
423:
412:
407:
404:
401:
397:
393:
380:
376:
369:
364:
360:
354:
349:
346:
343:
339:
330:
321:
315:
304:
288:
280:
276:
269:
264:
260:
254:
249:
246:
243:
239:
235:
229:
223:
215:
198:
192:
188:
184:
181:
178:
173:
169:
165:
160:
156:
151:
142:
138:
134:
130:
114:
108:
105:
102:
94:
90:
82:
80:
78:
74:
70:
66:
65:normed spaces
62:
58:
54:
50:
46:
42:
41:linear spaces
38:
34:
30:
19:
4877:Polynomially
4806:Grothendieck
4799:tame Fréchet
4749:Bornological
4609:Linear cone
4601:Convex cone
4576:Banach disks
4518:Sesquilinear
4490:
4373:Main results
4363:Vector space
4318:Completeness
4313:Banach space
4218:Balanced set
4192:Distribution
4130:Applications
3983:Krein–Milman
3968:Closed graph
3768:
3753:
3740:
3720:
3713:
3688:
3682:
3673:
3634:
3517:
3483:spaces with
3417:spaces with
3354:
3341:
3331:
3290:
3286:
3282:
3278:
3274:
3270:
3244:
3214:
3181:
3179:
3174:
3171:
3144:
3113:
2955:
2952:
2872:
2828:
2739:with domain
2678:
2673:
2663:
2654:
2648:
2614:
2604:exhibited a
2599:
2581:
2572:
2567:
2563:
2559:
2558:by defining
2555:
2450:
2415:
2409:
2405:
2397:
2393:
2391:
2386:
2301:
2297:
2293:
2289:
2285:
2281:
2277:
2273:
2272:Assume that
2210:
2202:
2198:
2196:
2188:
2183:
2170:is also not
2167:
2119:
2079:
2075:
2071:
2067:
2041:
2035:
2031:
2005:
1906:
1895:real numbers
1892:
1884:
1694:
1420:
1337:, that is,
1335:uniform norm
1308:
1195:
1185:
1181:
1177:
1173:
1169:
1165:
1161:
1160:exists. If
1157:
1150:
1148:
878:
761:
616:
213:
136:
132:
128:
92:
88:
86:
68:
26:
4871:Quasinormed
4784:FK-AK space
4678:Linear span
4673:Convex hull
4658:Affine hull
4461:Almost open
4401:Hahn–Banach
4147:Heat kernel
4137:Hardy space
4044:Trace class
3958:Hahn–Banach
3920:Topological
2643:, there is
2449:vectors in
2083:) = π, but
2065:. Then lim
1903:Hamel basis
1721:instead of
1476:defined on
1424:-at-a-point
61:dimensional
33:linear maps
29:mathematics
4986:Categories
4911:Stereotype
4769:(DF)-space
4764:Convenient
4503:Functional
4471:Continuous
4456:Linear map
4396:F. Riesz's
4338:Linear map
4080:C*-algebra
3895:Properties
3665:References
3526:given by
3344:dual space
3112:is called
2610:set theory
2595:set theory
2180:Vitali set
2172:measurable
2144:(not over
1824:dual space
1422:derivative
57:continuous
51:(that is,
4927:Uniformly
4886:Reflexive
4734:Barrelled
4730:Countably
4642:Symmetric
4540:Transpose
4054:Unbounded
4049:Transpose
4007:Operators
3936:Separable
3931:Reflexive
3916:Algebraic
3902:Barrelled
3547:∫
3540:‖
3534:‖
3524:quasinorm
3308:×
3221:from to
3195:×
3129:¯
3076:¯
3049:¯
3036:Γ
3007:×
2982:¯
2969:Γ
2933:×
2881:Γ
2842:
2809:→
2803:⊆
2791:
2750:
2538:…
2512:for each
2499:‖
2486:‖
2430:≥
2097:π
2053:π
1981:π
1952:→
1921:π
1899:rationals
1863:→
1838:∗
1748:→
1709:∞
1706:→
1681:∞
1678:→
1657:⋅
1641:
1580:≥
1532:
1365:∈
1351:‖
1345:‖
1294:‖
1281:‖
1273:‖
1251:‖
1082:ϵ
1038:δ
1003:ϵ
979:⊆
970:δ
924:ϵ
921:≤
918:δ
889:ϵ
865:‖
854:−
848:‖
842:≤
839:‖
825:−
813:‖
807:‖
787:−
772:‖
721:‖
715:‖
706:≤
658:∑
649:≤
646:‖
631:‖
615:for some
603:‖
597:‖
591:≤
551:∑
524:‖
502:‖
475:Letting
460:‖
438:‖
398:∑
394:≤
340:∑
328:‖
313:‖
240:∑
182:…
112:→
37:functions
4971:Category
4922:Strictly
4896:Schwartz
4836:LF-space
4831:LB-space
4789:FK-space
4759:Complete
4739:BK-space
4664:Relative
4611:(subset)
4603:(subset)
4530:Seminorm
4513:Bilinear
4256:Category
4068:Algebras
3950:Theorems
3907:Complete
3876:Schwartz
3822:glossary
3691:: 1–56,
3643:See also
3114:closable
2951:we call
2774:written
2402:sequence
2176:additive
1967:so that
1453:′
1155:supremum
944:so that
861:′
832:′
800:′
390:‖
335:‖
73:complete
4936:)
4884:)
4826:K-space
4811:Hilbert
4794:Fréchet
4779:F-space
4754:Brauner
4747:)
4732:)
4714:Asplund
4696:)
4666:)
4586:Bounded
4481:Compact
4466:Bounded
4403: (
4059:Unitary
4039:Nuclear
4024:Compact
4019:Bounded
4014:Adjoint
3988:Min–max
3881:Sobolev
3866:Nuclear
3856:Hilbert
3851:Fréchet
3816: (
3705:0265151
3273:() and
3145:closure
2637:F-space
4948:Webbed
4934:Quasi-
4856:Montel
4846:Mackey
4745:Ultra-
4724:Banach
4632:Radial
4596:Convex
4566:Affine
4508:Linear
4476:Closed
4300:(TVSs)
4034:Normal
3871:Orlicz
3861:Hölder
3841:Banach
3830:Spaces
3818:topics
3775:
3760:
3728:
3703:
3637:groups
3116:, and
2956:closed
2666:closed
2358:where
2213:where
1880:closed
736:Thus,
4906:Smith
4891:Riesz
4882:Semi-
4694:Quasi
4688:Polar
3846:Besov
2649:every
2606:model
2445:) of
2296:from
2284:from
2174:; an
756:is a
135:. If
4525:Norm
4449:form
4437:Maps
4194:(or
3912:Dual
3773:ISBN
3758:ISBN
3726:ISBN
3500:<
3494:<
3434:<
3428:<
3342:The
3243:and
2201:and
1569:for
1419:The
1168:and
1117:and
1053:and
892:>
91:and
87:Let
3693:doi
3184:of
3182:all
3147:of
3025:If
2996:in
2902:of
2839:Dom
2788:Dom
2747:Dom
2719:to
2608:of
2592:ZFC
2554:of
2392:If
2300:to
2288:to
2241:or
2205:be
1907:any
1802:on
1695:as
1638:cos
1529:sin
1358:sup
1184:to
1149:If
490:sup
305:,
212:in
139:is
131:to
27:In
4988::
3820:–
3701:MR
3699:,
3689:92
3285:↦
2625:BP
2623:+
2621:DC
2389:.
2106:0.
1882:.
1188:.
31:,
4932:(
4917:B
4915:(
4875:(
4743:(
4728:(
4692:(
4662:(
4412:)
4290:e
4283:t
4276:v
4198:)
3922:)
3918:/
3914:(
3824:)
3806:e
3799:t
3792:v
3779:.
3764:.
3735:.
3708:.
3695::
3620:.
3617:x
3614:d
3607:|
3603:)
3600:x
3597:(
3594:f
3590:|
3586:+
3583:1
3577:|
3573:)
3570:x
3567:(
3564:f
3560:|
3551:I
3543:=
3537:f
3503:1
3497:p
3491:0
3469:p
3465:L
3440:,
3437:1
3431:p
3425:0
3404:)
3401:x
3398:d
3395:,
3391:R
3387:(
3382:p
3378:L
3332:X
3314:,
3311:Y
3305:X
3291:x
3289:(
3287:p
3283:x
3279:T
3275:C
3271:C
3256:R
3245:Y
3230:R
3215:X
3201:.
3198:Y
3192:X
3175:X
3158:.
3155:T
3126:T
3100:T
3081:,
3073:T
3045:)
3042:T
3039:(
3013:.
3010:Y
3004:X
2978:)
2975:T
2972:(
2953:T
2939:,
2936:Y
2930:X
2910:T
2890:)
2887:T
2884:(
2854:.
2851:)
2848:T
2845:(
2829:X
2815:.
2812:Y
2806:X
2800:)
2797:T
2794:(
2785::
2782:T
2762:,
2759:)
2756:T
2753:(
2727:Y
2707:X
2687:T
2568:X
2564:T
2560:T
2556:X
2535:,
2532:2
2529:,
2526:1
2523:=
2520:n
2494:n
2490:e
2483:n
2480:=
2477:)
2472:n
2468:e
2464:(
2461:T
2451:X
2433:1
2427:n
2419:(
2416:n
2413:)
2410:n
2406:e
2404:(
2398:f
2394:X
2387:Y
2371:0
2367:y
2344:0
2340:y
2336:)
2333:x
2330:(
2327:f
2324:=
2321:)
2318:x
2315:(
2312:g
2302:Y
2298:X
2294:g
2290:K
2286:X
2282:f
2278:Y
2274:X
2260:.
2256:C
2252:=
2249:K
2228:R
2224:=
2221:K
2211:K
2203:Y
2199:X
2184:f
2168:f
2153:R
2131:Q
2120:f
2103:=
2100:)
2094:(
2091:f
2080:n
2076:r
2074:(
2072:f
2068:n
2042:n
2039:}
2036:n
2032:r
2017:R
2006:f
1993:,
1990:0
1987:=
1984:)
1978:(
1975:f
1955:R
1948:R
1944::
1941:f
1866:X
1860:X
1834:X
1810:X
1786:T
1766:0
1763:=
1760:)
1757:0
1754:(
1751:T
1745:)
1740:n
1736:f
1732:(
1729:T
1703:n
1675:n
1672:=
1667:n
1663:)
1660:0
1652:2
1648:n
1644:(
1633:2
1629:n
1622:=
1619:)
1614:n
1610:f
1606:(
1603:T
1583:1
1577:n
1555:n
1551:)
1548:x
1543:2
1539:n
1535:(
1523:=
1520:)
1517:x
1514:(
1509:n
1505:f
1484:X
1463:)
1460:0
1457:(
1450:f
1446:=
1443:)
1440:f
1437:(
1434:T
1407:.
1403:|
1399:)
1396:x
1393:(
1390:f
1386:|
1380:]
1377:1
1374:,
1371:0
1368:[
1362:x
1354:=
1348:f
1317:X
1289:i
1285:e
1277:/
1270:)
1265:i
1261:e
1257:(
1254:T
1231:T
1209:i
1205:e
1186:Y
1182:X
1178:Y
1174:X
1170:Y
1166:X
1162:Y
1158:M
1151:X
1134:)
1131:x
1128:(
1125:f
1105:x
1085:)
1079:,
1076:)
1073:x
1070:(
1067:f
1064:(
1061:B
1041:)
1035:,
1032:x
1029:(
1026:B
1018:(
1006:)
1000:,
997:)
994:x
991:(
988:f
985:(
982:B
976:)
973:)
967:,
964:x
961:(
958:B
955:(
952:f
932:K
928:/
898:,
895:0
879:K
858:x
851:x
845:K
836:)
829:x
822:x
819:(
816:f
810:=
804:)
797:x
793:(
790:f
784:)
781:x
778:(
775:f
762:f
744:f
724:.
718:x
712:M
709:C
703:M
699:)
694:|
688:i
684:x
679:|
673:n
668:1
665:=
662:i
653:(
643:)
640:x
637:(
634:f
617:C
600:x
594:C
587:|
581:i
577:x
572:|
566:n
561:1
558:=
555:i
530:,
527:}
521:)
516:i
512:e
508:(
505:f
499:{
494:i
486:=
483:M
463:.
457:)
452:i
448:e
444:(
441:f
434:|
428:i
424:x
419:|
413:n
408:1
405:=
402:i
386:)
381:i
377:e
373:(
370:f
365:i
361:x
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350:1
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344:i
331:=
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322:x
319:(
316:f
289:,
286:)
281:i
277:e
273:(
270:f
265:i
261:x
255:n
250:1
247:=
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236:=
233:)
230:x
227:(
224:f
214:X
199:)
193:n
189:e
185:,
179:,
174:2
170:e
166:,
161:1
157:e
152:(
137:X
133:Y
129:X
115:Y
109:X
106::
103:f
93:Y
89:X
20:)
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