Editing Banach–Alaoglu theorem (section)

==Sequential Banach–Alaoglu theorem==

A special case of the Banach–Alaoglu theorem is the sequential version of the theorem, which asserts that the closed unit ball of the dual space of a [[separable metric space|separable]] normed vector space is [[sequentially compact]] in the weak-* topology. 
In fact, the weak* topology on the closed unit ball of the dual of a separable space is [[metrizable]], and thus compactness and sequential compactness are equivalent.

Specifically, let <math>X</math> be a separable normed space and <math>B</math> the closed unit ball in <math>X^{\prime}.</math> Since <math>X</math> is separable, let <math>x_{\bull} = \left(x_n\right)_{n=1}^{\infty}</math> be a countable dense subset. 
Then the following defines a metric, where for any <math>x, y \in B</math> 
<math display=block>\rho(x,y) = \sum_{n=1}^\infty \, 2^{-n} \, \frac{\left|\langle x - y, x_n \rangle\right|}{1 + \left|\langle x - y, x_n \rangle\right|}</math>
in which <math>\langle\cdot, \cdot\rangle</math> denotes the duality pairing of <math>X^{\prime}</math> with <math>X.</math> 
Sequential compactness of <math>B</math> in this metric can be shown by a [[diagonalization argument]] similar to the one employed in the proof of the [[Arzelà–Ascoli theorem]].

Due to the constructive nature of its proof (as opposed to the general case, which is based on the axiom of choice), the sequential Banach–Alaoglu theorem is often used in the field of [[partial differential equations]] to construct solutions to PDE or [[Calculus of variations|variational problems]]. 
For instance, if one wants to minimize a functional <math>F : X^{\prime} \to \R</math> on the dual of a separable normed vector space <math>X,</math> one common strategy is to first construct a minimizing sequence <math>x_1, x_2, \ldots \in X^{\prime}</math> which approaches the infimum of <math>F,</math> use the sequential Banach–Alaoglu theorem to extract a subsequence that converges in the weak* topology to a limit <math>x,</math> and then establish that <math>x</math> is a minimizer of <math>F.</math> 
The last step often requires <math>F</math> to obey a (sequential) [[Semi-continuity|lower semi-continuity]] property in the weak* topology.

When <math>X^{\prime}</math> is the space of finite Radon measures on the real line (so that <math>X = C_0(\R)</math> is the space of continuous functions vanishing at infinity, by the [[Riesz–Markov–Kakutani representation theorem|Riesz representation theorem]]), the sequential Banach–Alaoglu theorem is equivalent to the [[Helly selection theorem]].

{{math proof|drop=hidden|proof=
For every <math>x \in X,</math> let
<math display=block>D_x = \{c \in \Complex : |c| \leq \|x\|\}</math>
and let
<math display=block>D = \prod_{x \in X} D_x</math>
be endowed with the [[product topology]]. 
Because every <math>D_x</math> is a compact subset of the complex plane, [[Tychonoff's theorem]] guarantees that their product <math>D</math> is compact. 

The closed unit ball in <math>X^{\prime},</math> denoted by <math>B_1^{\,\prime},</math> can be identified as a subset of <math>D</math> in a natural way:
<math display=block>\begin{alignat}{4}
F :\;&& B_1^{\,\prime} &&\;\to    \;& D \\[0.3ex]
     && f              &&\;\mapsto\;& (f(x))_{x \in X}. \\
\end{alignat}</math>

This map is injective and it is continuous when <math>B_1^{\,\prime}</math> has the [[weak-* topology]]. 
This map's inverse, defined on its image, is also continuous. 

It will now be shown that the image of the above map is closed, which will complete the proof of the theorem. 
Given a point <math>\lambda_{\bull} = \left(\lambda_x\right)_{x \in X} \in D</math> and a net <math>\left(f_i(x)\right)_{x \in X}</math> in the image of <math>F</math> indexed by <math>i \in I</math> such that 
<math display=block>\lim_{i} \left(f_i(x)\right)_{x \in X} \to \lambda_{\bull} \quad \text{ in } D,</math>
the functional <math>g : X \to \Complex</math> defined by
<math display=block>g(x) = \lambda_x \qquad \text{ for every } x \in X,</math>
lies in <math>B_1^{\,\prime}</math> and <math>F(g) = \lambda_{\bull}.</math> 
<math>\blacksquare</math>
}}