Prove: $M +N$ is closed in $B$ iff $∥m∥_B +∥n∥_B ≤ c∥m+n∥_B$, for all $m ∈ M$ and $n ∈ N$.

Question

Let $B$ be a Banach space and $M,N$ closed subspaces of $B$ such that $M ∩N = \{0\}$. Prove: $M +N$ is closed in $B$ iff $∥m∥_B +∥n∥_B ≤ c∥m+n∥_B$, for all $m ∈ M$ and $n ∈ N$.

My Work:

If $Y$ is a subspace of $B$ then $Y$ is complete iff $Y$ is closed (Theorem).

I proved the backward direction using this thorem. For the forward direction I was going to get a contradiction assuming for all $c$, there are some $m\in M$ and $n\in N$ such that $∥m∥_B +∥n∥_B > c∥m+n∥_B$ . But afterwards I was stuck. Can anybody please give me a hint?

Answer 1

Your idea of proving the remaining direction by contradiction seems to be a sensible approach.

A couple of hints to help you along:

Assuming $M+N$ is complete, the function $f:M+N\to \mathbb{R}: m+n\mapsto ||m||+||n||$ is continuous and well-defined (why?).

Now, take $(m_k,n_k)$ such that $||m_k||+||n_k||\geq k ||m_k+n_k||$.

What can you say about $f(m_k+n_k)$? Can you possibly force your $m_k,n_k$ to converge to a certain point, while $f(m_k+n_k)$ doesn't?