# nearest point and closed complement of a subspace in norm spaces

It is well-known that in any Hilbert space $H$, each closed subspace $Y$ admits a closed complement $Y^\perp$. This result also implies that there exists a best approximation point to $Y$ for any $x\in H$. In an arbitrary norm linear space $X$, if $Y$ is a finite dimensional subspace of $X$ then it also has a closed direct sum complement. I am trying to find two counterexamples

$(1)$ A closed subspace $Y$ in a norm space $X$ (not Hilbert) might not have nearest point for some points $x\notin Y$.

$(2)$ In a norm space $X$, if the assumption finite dimensional of $Y$ is invalid then $Y$ might not admit closed complement.

• Yes, $(2)$ is Ok. In $(1)$ I want to show that even subspaces (a very special case of convex subsets) also can not admit nearest point. – Richkent Mar 16 '14 at 16:40