## The Hahn-Banach theorem and two applicationsNovember 28, 2009

Posted by Akhil Mathew in analysis, functional analysis, MaBloWriMo.
Tags: , , , ,

I have been finishing my MaBloWriMo series on differential geometry with a proof of the Myers comparison theorem, which right now has only an outline, but will rely on the second variation formula for the energy integral.  After that, it looks like I’ll be posting somewhat more randomly.   Here I will try something different.

The Hahn-Banach theorem is a basic result in functional analysis, which simply states that one can extend a linear function from a subspace while preserving certain bounds, but whose applications are quite manifold.

Edit (12/5): This material doesn’t look so great on WordPress.  So, here’s the PDF version.  Note that the figure is omitted in the file.

The Hahn-Banach theorem

Theorem 1 (Hahn-Banach) Let ${X}$ be a vector space, ${g: X \rightarrow \mathbb{R}_{\geq 0}}$ a positive homogeneous (i.e. ${g(tx) = tg(x), t >0}$) and sublinear (i.e. ${g(x+y) \leq g(x) + g(y)}$) function.

Suppose ${Y}$ is a subspace and ${\lambda: Y \rightarrow \mathbb{R}}$ is a linear function with ${\lambda(y) \leq g(y)}$ for all ${y \in Y}$.

Then there is an extension of ${\lambda}$ to a functional ${\tilde{\lambda}: X \rightarrow \mathbb{R}}$ with

$\displaystyle \tilde{\lambda}(x) \leq g(x), \ x \in X.$

I’ll omit the proof; I want to discuss why it is so interesting.  One of its applications lies in questions of the form “are elements of this form dense in the space”? The reason is that if ${X}$ is a normed linear space and ${Y}$ a closed subspace, the quotient vector space ${X/Y}$ is a norm with the norm ${|x+Y| := \inf_{y\in Y} |x-y|.}$ (The closedness condition is necessary because otherwise there might be nonzero elements of ${X/Y}$ with zero norm.) (more…)