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wiki says The spaces $L_p([0, 1])$ for $0 < p < 1$ are equipped with the F-norm they are not locally convex, since the only convex neighborhood of zero is the whole space Why is this so?

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up vote 4 down vote accepted

Key fact. Given a function $f\in L^p([0,1])$ and a positive $\epsilon>0$, one can write $f$ as a finite convex combination of functions $g_1,\dots,g_n$ such that $\|g_k\|_{L^p}<\epsilon$ for all $k=1,\dots, n$.

Having the above, we conclude as follows: if $U$ is a convex neighborhood of $0$, then it contains the set $\{g: \|g\|_{L^p}<\epsilon\}$ for some $\epsilon>0$. The above fact, together with convexity, imply that $U$ contains all $L^p$ functions.

Proof of the key fact (sketch): For any $n$ there is a partition of $[0,1]$ into intervals $J_1,\dots,J_n$ such that $\int_{J_k} |f|^p=n^{-1}\int_0^1|f|^p$. Let $g_k=n\,f\,\chi_{J_k}$. Calculate $\|g_k\|_{L^p}$ and observe that it tends to $0$ as $n\to\infty$.

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thanks for the nice explanation – Koushik Dec 29 '12 at 7:33

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