Brauner space

In functional analysis and related areas of mathematics Brauner space is a complete compactly generated locally convex space $X$ having a sequence of compact sets $K_{n}$ such that every other compact set $T\subseteq X$ is contained in some $K_{n}$ .

Brauner spaces are named after Kalman Brauner,^[1] who first started to study them. All Brauner spaces are stereotype and are in the stereotype duality relations with Fréchet spaces:^[2]^[3]

for any Fréchet space $X$ its stereotype dual space^[4] $X^{\star }$ is a Brauner space,
and vice versa, for any Brauner space $X$ its stereotype dual space $X^{\star }$ is a Fréchet space.

Examples

Let $M$ be a $\sigma$ -compact locally compact topological space, and ${\mathcal {C}}(M)$ the space of all functions on $M$ (with values in ${\mathbb R}$ or ${{\mathbb C}}$ ), endowed with the usual topology of uniform convergence on compact sets in $M$ . The dual space ${\mathcal {C}}^{\star }(M)$ of measures with compact support in $M$ with the topology of uniform convergence on compact sets in ${\mathcal {C}}(M)$ is a Brauner space.
Let $M$ be a smooth manifold, and ${\mathcal {E}}(M)$ the space of smooth functions on $M$ (with values in ${\mathbb R}$ or ${{\mathbb C}}$ ), endowed with the usual topology of uniform convergence with each derivative on compact sets in $M$ . The dual space ${\mathcal {E}}^{\star }(M)$ of distributions with compact support in $M$ with the topology of uniform convergence on bounded sets in ${\mathcal {E}}(M)$ is a Brauner space.
Let $M$ be a Stein manifold and ${{\mathcal O}}(M)$ the space of holomorphic functions on $M$ with the usual topology of uniform convergence on compact sets in $M$ . The dual space ${\mathcal {O}}^{\star }(M)$ of analytic functionals on $M$ with the topology of uniform convergence on biunded sets in ${{\mathcal O}}(M)$ is a Brauner space.
Let $G$ be a compactly generated Stein group. The space ${\mathcal {O}}_{\exp }(G)$ of holomorphic functions of exponential type on $G$ is a Brauner space with respect to a natural topology.^[3]

Notes

↑ K.Brauner (1973).
↑ S.S.Akbarov (2003).
1 2 S.S.Akbarov (2009).
↑ 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

Schaefer, Helmuth H. (1966). Topological vector spaces. New York: The MacMillan Company. ISBN 0-387-98726-6.

Robertson, A.P.; Robertson, W.J. (1964). Topological vector spaces. Cambridge Tracts in Mathematics. 53. Cambridge University Press.
Brauner, K. (1973). "Duals of Frechet spaces and a generalization of the Banach-Dieudonne theorem". Duke Math. Jour. 40 (4): 845–855. doi:10.1215/S0012-7094-73-04078-7.
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. doi:10.1007/s10958-009-9646-1. (subscription required (help)).

Functional analysis

Set / subset types	Absolutely convex Absorbing Balanced Bounded Convex Radial Star-shaped Symmetric Linear cone (subset) Convex cone (subset)

TVS types	Banach Barrelled Bornological Brauner F-space Finite-dimensional Fréchet (tame) Hilbert LF-space Locally convex Mackey Montel Nuclear Normed (norm) Quasinormed Reflexive Riesz Smith Stereotype Webbed Topological tensor product (of Hilbert spaces)

Mapping topologies	Dual Dual space Operator Ultraweak Weak (operator) Mackey Strong (polar operator) Ultrastrong Uniform convergence

Linear operators	Adjoint Bilinear (form) Bounded / Unbounded Closed Continuous / Discontinuous Compact Fredholm Hilbert–Schmidt Functionals (positive) Normal Nuclear Self-adjoint Strictly singular Trace class Transpose Unitary

Set operations	Algebraic interior (core) Interior Minkowski addition Polar

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–Saks Banach–Mazur Bessel's inequality Cauchy–Schwarz inequality Closed range Closed graph Eberlein–Šmulian Freudenthal spectral Gelfand–Mazur Goldstine Hahn–Banach (hyperplane separation) Kakutani fixed-point Lomonosov's invariant subspace Mackey–Arens Mazur's lemma M. Riesz extension Open mapping Parseval's identity Schauder fixed-point

Analysis	Bochner space Differentiation in Fréchet spaces Derivatives (Fréchet Gâteaux functional holomorphic) Integrals (Bochner Dunford Gelfand–Pettis regulated weak) Functional calculus (Borel continuous holomorphic) Inverse function theorem (Nash–Moser theorem) Vector measure Weakly measurable function

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