[Math] Is projection on a closed subspace of a Banach space bounded

Question

We know that any projection on a closed subspace of a Hilbert space is bounded. Is it true for any Banach space?
Any help would be appreciable.

Answer 1

If the projection $P \colon E \to F$, where $E$ is Banach and $F$ a closed subspace of $E$, is continuous (bounded), then we have the decomposition

$$E \cong \ker P \oplus F.$$

Thus a necessary condition for the existence of a continuous projection onto a closed subspace $F$ is that $F$ is complemented. That condition is of course also sufficient, if $E \cong F \oplus G$ with a closed subspace $G$, the projection along $G$ is continuous.

Not all closed subspaces in a Banach space are complemented, in general, hence in general, a continuous projection onto $F$ need not exist.