How to show that if $X=M \oplus N$ is a Banach space then $\exists c > 0$ $\forall m\in M \, \forall n\in N: ||m||+||n|| \leq c ||m+n||$?

Question

Let $X=M \oplus N$, where $X$ is a Banach space and $M$ and $N$ are closed subspaces of $X$. How to prove that $\exists c > 0$ constant such that $\forall m \in M$ and $\forall n \in N$ $$ \left\lVert m\right\rVert + \left\lVert n\right\rVert \leq c \left\lVert m + n\right\rVert$$?

Answer 1

Try showing boundedness of the projection operators $P(m+n) = m$ and $Q(m+n) = n$ by the use of, for instance, closed graph theorem. This will show that $\|m\| \leq c_1\|m + n\|$ and $\|n\| \leq c_2\|m+n\|$.