How to show that the projections onto the summands in a direct sum decomposition of a Banach space have the same Lipchitz constant?

Question

Suppose that $X$ is a Banach space and it's the direct sum of the closed subspaces $Y$ and $Z$, i.e, for any $x\in X$, there's a unique decomposition $x=y+z$ with $y\in Y$ and $z\in Z$. I'm trying to show that $||y||\leq C||x||$ and $||z||\leq C||x||$ for the same constant $C$, $\forall x\in X$.

I'm at a complete loss about which result to use and how to proceed. This is an old qualifying exam problem and I'd appreciate some leads. Thank you.

Answer 1

Doesn't this just follow from the fact that projection maps are continuous?

Answer 2

Let $P:X \to X$ denote the projection on $Y$, hence $P(y+z)=y$. Since $Y$ and $Z$ are closed, the closed graph theorem gives that $P$ is bounded.

Hence: $||y||=||Px|| \le ||P||*||x|$

Let $Q:I-P$. Then a similar argument gives $||z||=||Qx|| \le ||Q||*||x|$.

Take $C:= \max\{||P||,||Q||\}$