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.
Thanks in advance!