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I am just wondering, how to prove the Hahn-Banach theorem constructively for a finite dimensional normed vector space.

Thanks in advance for any helpful answers.

edited Sep 27 2011 at 19:40

asked Sep 27 2011 at 19:32

q.g
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I am not sure this question reflects the level of this site. I am a bit puzzled about the attention it gets. – András Bátkai Sep 27 2011 at 21:55

@Andras: IMHO, the question might be borderline, but it is definitely not trivial (or at least, it is as non-trivial as the usual proof of the Hahn-Banach theorem, where the inductive step is actually a bit delicate). – Yemon Choi Sep 28 2011 at 1:00

I believe that constructive versions of the Hahn-Banach theorem have been studied in several papers by Thierry Coquand. You might look at, e.g., his paper "Geometric Hahn-Banach Theorem" in Math. Proc. Camb. Phil. Soc. (2006), 140, 313. – Michael A Warren Sep 28 2011 at 2:40

I was confused because this is usually optional homework on my beginning FA course. – András Bátkai Sep 28 2011 at 7:42

Igor Rivin · Accepted Answer · 2011-09-27 19:41:23Z UTC

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Same way as for the infinite dimensional case, except you avoid Zorn's lemma by counting dimensions.

answered Sep 27 2011 at 19:41

Igor Rivin
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Isn't the whole thing trivial as soon as you know that your subspace is complemented? – András Bátkai Sep 27 2011 at 20:52

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Preserving norms isn't trivial. – Benjamin Hayes Sep 27 2011 at 20:54

Nikita Sidorov · Answer 2 · 2011-09-27 21:34:22Z UTC

The idea is to show that one can extend a linear functional from an $n$-dimensional space to a space of dimension $n+1$ without increasing its norm. See, for instance, my notes (Lemma E.2)

In fact, by doing so, you can prove THBT constructively for any separable space.

How to prove the Hahn-Banach constructively

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