Polar topology

In functional analysis and related areas of mathematics a polar topology, topology of ${\mathcal {A}}$ -convergence or topology of uniform convergence on the sets of ${\mathcal {A}}$ is a method to define locally convex topologies on the vector spaces of a dual pair.

Definitions

Let $(X,Y,\langle ,\rangle )$ be a dual pair of vector spaces $X$ and $Y$ over the field $\mathbb {F}$ , either the real or complex numbers.

A set $A\subseteq X$ is said to be bounded in $X$ with respect to $Y$ , if for each element $y\in Y$ the set of values $\{\langle x,y\rangle ;x\in A\}$ is bounded:

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

This condition is equivalent to the requirement that the polar $A^{\circ }$ of the set $A$ in $Y$

A^{\circ }=\{y\in Y:\quad \sup _{{x\in A}}|\langle x,y\rangle |\leq 1\}

is an absorbent set in $Y$ , i.e.

\bigcup _{{\lambda \in {{\mathbb F}}}}\lambda \cdot A^{\circ }=Y.

Let now ${\mathcal {A}}$ be a family of bounded sets in $X$ (with respect to $Y$ ) with the following properties:

each point $x$ of $X$ belongs to some set $A\in{\mathcal A}$

\forall x\in X\qquad \exists A\in {{\mathcal A}}\qquad x\in A,

each two sets $A\in{\mathcal A}$ and $B\in{\mathcal A}$ are contained in some set $C\in{\mathcal A}$ :

\forall A,B\in {\mathcal A}\qquad \exists C\in {\mathcal A}\qquad A\cup B\subseteq C,

${\mathcal A}$ is closed under the operation of multiplication by scalars:

\forall A\in {{\mathcal A}}\qquad \forall \lambda \in {{\mathbb F}}\qquad \lambda \cdot A\in {{\mathcal A}}.

Then the seminorms of the form

\|y\|_{A}=\sup _{{x\in A}}|\langle x,y\rangle |,\qquad A\in {{\mathcal A}},

define a Hausdorff locally convex topology on $Y$ which is called the polar topology^[1] on $Y$ generated by the family of sets ${\mathcal A}$ . The sets

U_{{B}}=\{x\in V:\quad \|\varphi \|_{B}<1\},\qquad B\in {{\mathcal B}},

form a local base of this topology. A net of elements $y_{i}\in Y$ tends to an element $y\in Y$ in this topology if and only if

\forall A\in {{\mathcal A}}\qquad \|y_{i}-y\|_{A}=\sup _{{x\in A}}|\langle x,y_{i}\rangle -\langle x,y\rangle |{\underset {i\to \infty }{\longrightarrow }}0.

Because of this the polar topology is often called the topology of uniform convergence on the sets of ${\mathcal {A}}$ . The semi norm $\|y\|_{A}$ is the gauge of the polar set $A^{\circ }$ .

Examples

if ${\mathcal {A}}$ is the family of all bounded sets in $X$ then the polar topology on $Y$ coincides with the strong topology,
if ${\mathcal {A}}$ is the family of all finite sets in $X$ then the polar topology on $Y$ coincides with the weak topology,
the topology of an arbitrary locally convex space $X$ can be described as the polar topology defined on $X$ by the family ${\mathcal {A}}$ of all equicontinuous sets $A\subseteq X'$ in the dual space $X'$ .^[2]

Notes

↑ A.P.Robertson, W.Robertson (1964, III.2)
↑ In other words, $A\in{\mathcal A}$ iff $A\subseteq X'$ and there is a neighbourhood of zero $U\subseteq X$ such that $\sup _{{x\in U,f\in A}}|f(x)|<\infty$

References

Robertson, A.P.; Robertson, W. (1964). Topological vector spaces. Cambridge University Press.

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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Polar topology

Definitions

Examples

See also

Notes

References