Given an infinite dimensional normed vector space $X$, $S$ is a convex subset of $X$ with empty interior. Is it true that $S$ lies in a proper affine subspace of $X$?
There is a similar question but it only covers the finite dimensional case.
Convex set with empty interior lies in a proper affine subspace – infinite dimension
convex-analysisfunctional-analysis
Best Answer
No. Take $X=L^2(0,1)$ and $S=\{u\in X:\ u\ge 0 \ a.e.\}$. Then $S$ has no interior points, but $X=S-S$. Every $L^2$-function can be written as the difference of two non-negative $L^2$-functions.