Smith space

In functional analysis and related areas of mathematics, a Smith space is a complete compactly generated locally convex topological vector space $X$ having a universal compact set, i.e. a compact set $K$ which absorbs every other compact set $T \subseteq X$ (i.e. $T \subseteq λ \cdot K$ for some $λ > 0$ ).

Smith spaces are named after Marianne Ruth Freundlich Smith, who introduced them as duals to Banach spaces in some versions of duality theory for topological vector spaces. All Smith spaces are stereotype and are in the stereotype duality relations with Banach spaces:

for any Banach space $X$ its stereotype dual space $X^{⋆}$ is a Smith space,

and vice versa, for any Smith space $X$ its stereotype dual space $X^{⋆}$ is a Banach space.

Smith spaces are special cases of Brauner spaces.

Examples

As follows from the duality theorems, for any Banach space $X$ its stereotype dual space $X^{⋆}$ is a Smith space. The polar $K = B^{\circ}$ of the unit ball $B$ in $X$ is the universal compact set in $X^{⋆}$ . If $X^{*}$ denotes the normed dual space for $X$ , and $X^{'}$ the space $X^{*}$ endowed with the $X$ -weak topology, then the topology of $X^{⋆}$ lies between the topology of $X^{*}$ and the topology of $X^{'}$ , so there are natural (linear continuous) bijections

X^{*} \to X^{⋆} \to X^{'} .

If

X

is infinite-dimensional, then no two of these topologies coincide. At the same time, for infinite dimensional

X

the space

X^{⋆}

is not barreled (and even is not a Mackey space if

X

is reflexive as a Banach space).

If $K$ is a convex balanced compact set in a locally convex space $Y$ , then its linear span $ℂ K = span (K)$ possesses a unique structure of a Smith space with $K$ as the universal compact set (and with the same topology on $K$ ).
If $M$ is a (Hausdorff) compact topological space, and $𝒞 (M)$ the Banach space of continuous functions on $M$ (with the usual sup-norm), then the stereotype dual space $𝒞^{⋆} (M)$ (of Radon measures on $M$ with the topology of uniform convergence on compact sets in $𝒞 (M)$ ) is a Smith space. In the special case when $M = G$ is endowed with a structure of a topological group the space $𝒞^{⋆} (G)$ becomes a natural example of a stereotype group algebra.
A Banach space $X$ is a Smith space if and only if $X$ is finite-dimensional.

Notes

↑ Smith 1952.
↑ Akbarov 2003, p. 220.
↑ Akbarov 2009, p. 467.
↑ The stereotype dual space to a locally convex space $X$ is the space $X^{⋆}$ of all linear continuous functionals $f : X \to ℂ$ endowed with the topology of uniform convergence on totally bounded sets in $X$ .
↑ Akbarov 2003, p. 221, Example 4.8.
↑ Akbarov 2009, p. 468.
↑ Akbarov 2003, p. 272.

References

Smith, M.F. (1952). "The Pontrjagin duality theorem in linear spaces". Annals of Mathematics. 56 (2): 248–253. doi:10.2307/1969798. JSTOR 1969798.
Akbarov, S.S. (2003). "Pontryagin duality in the theory of topological vector spaces and in topological algebra". Journal of Mathematical Sciences. 113 (2): 179–349. doi:10.1023/A:1020929201133. S2CID 115297067.
Akbarov, S.S. (2009). "Holomorphic functions of exponential type and duality for Stein groups with algebraic connected component of identity". Journal of Mathematical Sciences. 162 (4): 459–586. arXiv:0806.3205. doi:10.1007/s10958-009-9646-1. S2CID 115153766.
Furber, R.W.J. (2017). Categorical Duality in Probability and Quantum Foundations (PDF) (PhD). Radboud University.

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[FOOTNOTESmith1952-1] Smith 1952.

[FOOTNOTEAkbarov2003220-2] Akbarov 2003, p. 220.

[FOOTNOTEAkbarov2009467-3] Akbarov 2009, p. 467.

[4] The stereotype dual space to a locally convex space $X$ is the space $X^{⋆}$ of all linear continuous functionals $f : X \to ℂ$ endowed with the topology of uniform convergence on totally bounded sets in $X$ .

[FOOTNOTEAkbarov2003221Example_4.8-5] Akbarov 2003, p. 221, Example 4.8.

[FOOTNOTEAkbarov2009468-6] Akbarov 2009, p. 468.

[FOOTNOTEAkbarov2003272-7] Akbarov 2003, p. 272.

v t e Topological vector spaces (TVSs)
Basic concepts	Banach space Completeness Continuous linear operator Linear functional Fréchet space Linear map Locally convex space Metrizability Operator topologies Topological vector space Vector space
Main results	Anderson–Kadec Banach–Alaoglu Closed graph theorem F. Riesz's Hahn–Banach (hyperplane separation Vector-valued Hahn–Banach) Open mapping (Banach–Schauder) Bounded inverse Uniform boundedness (Banach–Steinhaus)
Maps	Bilinear operator form Linear map Almost open Bounded Continuous Closed Compact Densely defined Discontinuous Topological homomorphism Functional Linear Bilinear Sesquilinear Norm Seminorm Sublinear function Transpose
Types of sets	Absolutely convex/disk Absorbing/Radial Affine Balanced/Circled Banach disks Bounding points Bounded Complemented subspace Convex Convex cone (subset) Linear cone (subset) Extreme point Pre-compact/Totally bounded Prevalent/Shy Radial Radially convex/Star-shaped Symmetric
Set operations	Affine hull (Relative) Algebraic interior (core) Convex hull Linear span Minkowski addition Polar (Quasi) Relative interior
Types of TVSs	Asplund B-complete/Ptak Banach (Countably) Barrelled BK-space (Ultra-) Bornological Brauner Complete Convenient (DF)-space Distinguished F-space FK-AK space FK-space Fréchet tame Fréchet Grothendieck Hilbert Infrabarreled Interpolation space K-space LB-space LF-space Locally convex space Mackey (Pseudo)Metrizable Montel Nuclear Quasibarrelled Quasi-complete Quasinormed (Polynomially Semi-) Reflexive Riesz Schwartz Semi-complete Smith Stereotype (B Strictly Uniformly) convex (Quasi-) Ultrabarrelled Uniformly smooth Webbed With the approximation property
Category

Smith space

Examples

See also

Notes

References