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.
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4$\begingroup$ I am not sure this question reflects the level of this site. I am a bit puzzled about the attention it gets. $\endgroup$– András BátkaiCommented Sep 27, 2011 at 21:55
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1$\begingroup$ @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). $\endgroup$– Yemon ChoiCommented Sep 28, 2011 at 1:00
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$\begingroup$ 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. $\endgroup$– Michael A WarrenCommented Sep 28, 2011 at 2:40
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$\begingroup$ I was confused because this is usually optional homework on my beginning FA course. $\endgroup$– András BátkaiCommented Sep 28, 2011 at 7:42
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2 Answers
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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
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$\begingroup$ Isn't the whole thing trivial as soon as you know that your subspace is complemented? $\endgroup$ Commented Sep 27, 2011 at 20:52
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2$\begingroup$ Preserving norms isn't trivial. $\endgroup$ Commented Sep 27, 2011 at 20:54
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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.
answered Sep 27, 2011 at 21:34