Editing Banach space (section)

====Weak topologies====

The ''[[weak topology]]'' on a Banach space <math>X</math> is the [[Comparison of topologies|coarsest topology]] on <math>X</math> for which all elements <math>x'</math> in the continuous dual space <math>X'</math> are continuous. 
The norm topology is therefore [[Comparison of topologies|finer]] than the weak topology. 
It follows from the Hahn–Banach separation theorem that the weak topology is [[Hausdorff space|Hausdorff]], and that a norm-closed [[Convex set|convex subset]] of a Banach space is also weakly closed.<ref>Theorem&nbsp;2.5.16, p.&nbsp;216 in {{harvtxt|Megginson|1998}}.</ref> 
A norm-continuous linear map between two Banach spaces <math>X</math> and <math>Y</math> is also ''weakly continuous'', that is, continuous from the weak topology of <math>X</math> to that of <math>Y.</math><ref>see II.A.8, p.&nbsp;29 in {{harvtxt|Wojtaszczyk|1991}}</ref>

If <math>X</math> is infinite-dimensional, there exist linear maps which are not continuous. The space <math>X^*</math> of all linear maps from <math>X</math> to the underlying field <math>\mathbb{K}</math> (this space <math>X^*</math> is called the [[Dual space#Algebraic dual space|algebraic dual space]], to distinguish it from <math>X'</math> also induces a topology on <math>X</math> which is [[finer topology|finer]] than the weak topology, and much less used in functional analysis.

On a dual space <math>X',</math> there is a topology weaker than the weak topology of <math>X',</math> called the ''[[weak topology|weak* topology]]''. 
It is the coarsest topology on <math>X'</math> for which all evaluation maps <math>x' \in X' \mapsto x'(x),</math> where <math>x</math> ranges over <math>X,</math> are continuous. 
Its importance comes from the [[Banach–Alaoglu theorem]].

{{math theorem|name=[[Banach–Alaoglu theorem]]|math_statement=Let <math>X</math> be a [[normed vector space]]. Then the [[Closed set|closed]] [[Ball (mathematics)|unit ball]] <math>B = \{x \in X \mid \|x\| \leq 1\}</math> of the dual space is [[Compact space|compact]] in the weak* topology.}}

The Banach–Alaoglu theorem can be proved using [[Tychonoff's theorem]] about infinite products of compact Hausdorff spaces. 
When <math>X</math> is separable, the unit ball <math>B'</math> of the dual is a [[Metrizable space|metrizable]] compact in the weak* topology.<ref name="DualBall">see Theorem&nbsp;2.6.23, p.&nbsp;231 in {{harvtxt|Megginson|1998}}.</ref>