I’ve been studying the Hahn – Banach theorem and I found this post very informative! ðŸ˜€

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.Â After that, it looks like Iâ€™ll be posting somewhat more randomly.Â Â 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 doesnâ€™t look so great on WordPress.Â So, hereâ€™s the PDF version.Â Note that the figure is omitted in the file.

**The Hahn-Banach theorem****Â Â Â Â Â Â Â Â Â Â Â Â Â **

**Theorem 1 (Hahn-Banach)***Let $latex {X}&fg=000000$ be a vector space, **$latex {g: X \rightarrow \mathbb{R}_{\geq 0}}&fg=000000$ a positive homogeneous (i.e. $latex {g(tx) = tg(x), t >0}&fg=000000$) and sublinear (i.e. $latex {g(x+y) \leq g(x) +â€¦*

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This theorem looks very abstract, but most of the theory of numerical resolution of PDEs is based on it. I find impressive the way our predecessors made such abstract thoughts on such “applied” problems, but I guess that’s what differentiate mathematicians from engineers, our job is to abstract ourselves!