# local convexity of $L_p$ spaces

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? http://en.wikipedia.org/wiki/Lp_space

#### Solutions Collecting From Web of "local convexity of $L_p$ spaces"

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 $n\to\infty$.