I am working on the following question:
$X$ is Banach. $A, B$ are closed subspaces such that $x \in X$ can be uniquely represented as the sum $x = a + b, a \in A , b \in B$. Show that $\exists k \in \mathbb{R}: \| a\| \leq k \|x\| \\ \| b\| \leq k \|x\| \hspace{2cm} \forall x \in X$.
I am completely stuck. I can't figure out how to use the closedness other than to conclude completeness of the space. Or perhaps to conclude that the projection maps will be bounded. But I can't proceed. Any nudge would be helpful.
