When is $M+N$ closed

Question

Let $X$ be a Banach space and $M,N$ be closed subspaces. If the range of linear transformation $x\to (x+M)\oplus (x+N)$ from $X$ into $X/M\oplus X/N$ is closed show that $M+N$ is closed.

or using $M^\perp+ N^\perp $ is norm closed to show $M+N$ is closed

Answer 1

I will give you a hint:

Let us start with the definition, i.e. take sequences $(x_n)_n$ in $M$ and $(y_n)_n$ in $N$ with $x_n + y_n \to z$ for some $z \in X$. We want to show $z \in M+N$.

Now, let us write $\Gamma : X \to X/M \oplus X/N, x \mapsto (x+M) \oplus (x+N)$. We have

$$ \Gamma(x_n) = (0, x_n +N) = (0, x_n+y_n+N) \to (0, z + N). $$

Why/how does the assumption help you now?