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

*Posted by Akhil Mathew in analysis, functional analysis, MaBloWriMo.*

Tags: convex sets, Hahn-Banach theorem, hyperplane separation theorem, linear functionals, Muntz approximation theorem

5 comments

Tags: convex sets, Hahn-Banach theorem, hyperplane separation theorem, linear functionals, Muntz approximation theorem

5 comments

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 *

**closed**subspace, the quotient vector space is a norm with the norm (The closedness condition is necessary because otherwise there might be nonzero elements of with zero norm.)

*(more…)*