Strong topology (polar topology)

In functional analysis and related areas of mathematics the strong topology is the finest polar topology, the topology with the most open sets, on a dual pair. The coarsest polar topology is called weak topology.

Definition

Let $(X,Y,\langle ,\rangle )$ be a dual pair of vector spaces over the field ${{\mathbb F}}$ of real ( ${\mathbb R}$ ) or complex ( ${{\mathbb C}}$ ) numbers. Let us denote by ${{\mathcal B}}$ the system of all subsets $B\subseteq X$ bounded by elements of $Y$ in the following sense:

\forall y\in Y\qquad \sup _{{x\in B}}|\langle x,y\rangle |<\infty .

Then the strong topology $\beta (Y,X)$ on $Y$ is defined as the locally convex topology on $Y$ generated by the seminorms of the form

||y||_{B}=\sup _{{x\in B}}|\langle x,y\rangle |,\qquad y\in Y,\qquad B\in {{\mathcal B}}.

In the special case when $X$ is a locally convex space, the strong topology on the (continuous) dual space $X'$ (i.e. on the space of all continuous linear functionals $f:X\to {{\mathbb F}}$ ) is defined as the strong topology $\beta (X',X)$ , and it coincides with the topology of uniform convergence on bounded sets in $X$ , i.e. with the topology on $X'$ generated by the seminorms of the form

||f||_{B}=\sup _{{x\in B}}|f(x)|,\qquad f\in X',

where $B$ runs over the family of all bounded sets in $X$ . The space $X'$ with this topology is called strong dual space of the space $X$ and is denoted by $X'_{\beta }$ .

Examples

If $X$ is a normed vector space, then its (continuous) dual space $X'$ with the strong topology coincides with the Banach dual space $X'$ , i.e. with the space $X'$ with the topology induced by the operator norm. Conversely $\beta(X, X')$ -topology on $X$ is identical to the topology induced by the norm on $X$ .

Properties

If $X$ is a barrelled space, then its topology coincides with the strong topology $\beta (X,X')$ on $X$ and with the Mackey topology on $X$ generated by the pairing $(X,X')$ .

References

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

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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