Let $(X, \|\cdot\|)$ be a Banach space, $Y \subset X$ a closed subspace. $Y$ is called complemented if there is a closed subspace $Z \subset X$ such that $X =Y \oplus Z$ as topological vector spaces.
Show that if a closed subspace $Y$ is not complemented, then the identity map on $Y$ can not be extended to a continuous linear function on $X$.
Hi, I think I should do this by contradiction, but I don't know how, I wish you could help me! Thank you so much!