Smith space

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

Smith spaces are named after Marianne Ruth Freundlich Smith, who introduced them^[1] 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:^[2]^[3]

for any Banach space $X$ its stereotype dual space^[4] $X^{\star }$ is a Smith space,

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

Notes

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

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.
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. (Subscription required (help)).
Furber, R.W.J. (2017). Categorical Duality in Probability and Quantum Foundations (PhD). Radboud University.

This article is issued from Wikipedia. The text is licensed under Creative Commons - Attribution - Sharealike. Additional terms may apply for the media files.

[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 $X^{\star }$ of all linear continuous functionals $f:X\to {\mathbb {C}}$ endowed with the topology of uniform convergence on totally bounded sets in $X$ .

Functional analysis (topics)
TVS types	Banach Banach Lattice Barrelled Bornological Brauner F-space Finite-dimensional Fréchet (tame) Hilbert (pre-Hilbert Polarization identity) LF-space Locally convex (Seminorms/Minkowski functionals) Mackey Montel Nuclear Normed (norm) Quasinormed Reflexive Riesz Smith Stereotype Strictly convex Webbed Topological tensor product (of Hilbert spaces)
Mapping topologies	Dual Dual space (Dual norm) Operator Ultraweak Weak (polar operator) Mackey Strong (polar operator) Ultrastrong Uniform convergence
Linear operators	Adjoint Bilinear (form operator sesquilinear) (Un)Bounded Closed Compact (Dis)Continuous Densely defined Fredholm Hilbert–Schmidt Functionals (positive) Normal Nuclear Self-adjoint Strictly singular Trace class Transpose Unitary
Operator theory	Banach algebras C-algebras Spectrum (C-algebra radius) Spectral theory (of ODEs Spectral theorem) Polar decomposition Singular value decomposition
Theorems	Banach–Alaoglu Banach–Mazur Banach–Saks Bessel's inequality Cauchy–Schwarz inequality Closed graph Closed range Eberlein–Šmulian Freudenthal spectral Gelfand–Mazur Gelfand–Naimark Goldstine Hahn–Banach (hyperplane separation) Kakutani fixed-point Krein–Milman Lomonosov's invariant subspace Mackey–Arens Mazur's lemma M. Riesz extension Riesz representation Open mapping Parseval's identity Schauder fixed-point
Analysis	Abstract Wiener space Bochner space Differentiation in Fréchet spaces Derivatives (Fréchet Gâteaux functional holomorphic) Integrals (Bochner Dunford Gelfand–Pettis regulated Paley–Wiener weak) Functional calculus (Borel continuous holomorphic) Inverse function theorem (Nash–Moser theorem) Measures (Lebesgue Projection-valued Vector) Weakly measurable function
Types of sets	Absolutely convex Absorbing Balanced Bounded Convex Convex cone (subset) Linear cone (subset) Radial Star-shaped Symmetric Zonotope
Subsets / set operations	Algebraic interior (core) Bounding points Convex hull Extreme point Interior Minkowski addition Polar

Smith space

See also

Notes

References