The Hahn-Banach theorem and two applications November 28, 2009Posted by Akhil Mathew in analysis, functional analysis, MaBloWriMo.
Tags: convex sets, Hahn-Banach theorem, hyperplane separation theorem, linear functionals, Muntz approximation theorem
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 be a vector space, a positive homogeneous (i.e. ) and sublinear (i.e. ) function.
Suppose is a subspace and is a linear function with for all .
Then there is an extension of to a functional with
A theorem of Mazur-Ulam on isometric maps of vector spaces November 22, 2009Posted by Akhil Mathew in analysis, functional analysis, MaBloWriMo.
Tags: isometries, linear maps
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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 of a normed linear space onto another normed linear space with is linear. (more…)