I know the Hahn Banach Theorem, Open Mapping Theorem and the Closed Graph Theorem. I think I need to use the Closed Graph Theorem on this problem but i'm not sure.
Let $X$ be a Banach space, and suppose that $Y$ and $Z$ are closed linear subspaces of $X$ such that $X$ is the direct sum of $Y$ and $Z$: $X=Y\oplus Z$ (that is, each $x\in X$ has a unique representation of the form $x=y+z,$ with $y\in Y$ and $z\in Z$). Prove that there exists a positive constant $C$ such that $$||y||\leq C||x||\text{ and }||z||\leq C||x||,$$ whenever $x=y+z,$ with $y\in Y$ and $z\in Z$.
This is a prelim prep problem and i'm not sure where to go. A complete solution would be preferred so I can walk myself through it. Thank you so much!
It is enough to show that the map defined by $p(x)=y$ is continuous, use the closed graph theorem since continuous linear functions are bounded:
If $(x_n=y_n+z_n)$ converges towards $(x,y)$, $y_n$ converges towards $y\in Y$ and $x_n-y_n=z_n\in Z$ converges towards $z$ implies that $p(x)=y$.