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The Hahn-Banach theorem and two applications November 28, 2009

Posted by Akhil Mathew in analysis, functional analysis, MaBloWriMo.
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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…)

A theorem of Mazur-Ulam on isometric maps of vector spaces November 22, 2009

Posted by Akhil Mathew in analysis, functional analysis, MaBloWriMo.
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I first posted this entry at Climbing Mount Bourbaki, where I have continued the MaBloWriMo series into topics in Riemannian geometry such as the Cartan-Hadamard theorem.  This particular material came up as part of the proof that distance-preserving maps between Riemannian manifolds are actually isometries.  However, the style of the entry seemed appropriate for this blog, so I’m placing it here as well.

The result in question is:

Theorem 1 (Mazur-Ulam) An isometry {M: X \rightarrow X'} of a normed linear space {X} onto another normed linear space {X'} with {M(0)=0} is linear. (more…)

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