TITLE: An infinite dimensional normed linear space is the union of two disjoint convex sets
QUESTION [6 upvotes]: Let $X$ be an infinite dimensional normed linear space. I want to show that there exist two disjoint convex sets $C_1$ and $C_2$ such that $X=C_1\cup C_2$ and both $C_1$ and $C_2$ are dense in $X$.
I proceed as follows: Since $X$ is infinite dimensional, there exists a discontinuous linear functional $f:X\to \mathbb K$. Then ker$f$ is dense in $X$. If we consider $C_1=\{x\in X:f(x)<0\}$ and $C_2=\{x\in X:f(x)\geq 0\}$, then $X=C_1\cup C_2$ and $C_1\cap C_2=\emptyset$. Since $C_2$ contains ker$f$, therefore, $C_2$ is dense. Also $C_1$ and $C_2$ are convex sets. But how to show that $C_1$ is dense?

REPLY [3 votes]: Since $f$ is not the trivial functional it is surjective. In particular, there exists $v$ such that $f(v)=-1$. Now $C_1$ contains the $v$-translation of $\ker f$: 
$$
v+\ker f\subset C_1,$$
and the left hand side is dense as it is the translation of a dense set.