Let $V$ be a Banach space and $X$ be a closed subspace of $V$. When $X$ is finite dimensional, the existence of its closed complement, that is, some closed subspace $Y$ such that $X\oplus Y=V$, follows from Hahn-Banach theorem.
What if $X$ is infinite dimensional? Is there a counter example?
Best Answer
As was pointed out by Kavi Rama Murthy, there is a theorem, already $50$ years old by now, proved by Lindenstrauss and Tzafriri, asserting that a Banach space $X$ is isomorphic to Hilbert space if and only if every closed subspace of $X$ is complemented. Therefore, every infinite dimensional Banach space not isomorphic to Hilbert space, contains uncomplemented subspaces. For a survey on the topic, see here: A survey on the complemented subspace problem.