# Weak topology (polar topology)

{{ safesubst:#invoke:Unsubst||$N=Unreferenced |date=__DATE__ |$B= {{#invoke:Message box|ambox}} }} In functional analysis and related areas of mathematics the weak topology is the coarsest polar topology, the topology with the fewest open sets, on a dual pair. The finest polar topology is called strong topology.

Under the weak topology the bounded sets coincide with the relatively compact sets which leads to the important Bourbaki–Alaoglu theorem.

## Definition

Given a dual pair $\displaystyle (X,Y,\langle ,\rangle )$ the weak topology Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): \sigma (X,Y) is the weakest polar topology on $\displaystyle X$ so that

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That is the continuous dual of Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): (X,\sigma (X,Y)) is equal to $\displaystyle Y$ up to isomorphism.

The weak topology is constructed as follows:

For every $\displaystyle y$ in $\displaystyle Y$ on $\displaystyle X$ we define a semi norm on $\displaystyle X$

$y}:X\to {\mathbb {R}$

with

$y$

This family of semi norms defines a locally convex topology on $\displaystyle X$ .

## Examples

• Given a normed vector space $\displaystyle X$ and its continuous dual $\displaystyle X'$ , Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): \sigma (X,X') is called the weak topology on $\displaystyle X$ and $\displaystyle \sigma (X',X)$ the weak* topology on $\displaystyle X'$