Sum of subspaces is closed iff inclination is positive

Question

It is a well-known result in functional analysis that the sum $M+N$ of two subspaces of a Banach space with $M\cap N=0$ is closed if and only if the inclination $$\widehat{(M,N)} := \inf_{x\in M, \|x\|=1} d(x,N)$$ is positive, i.e. $$ M+N \text{closed} \Leftrightarrow \widehat{(M,N)}>0.$$ Typically this is quoted from

Gurariĭ, V. I.: Openings and inclinations of subspaces of a Banach space. Teor. Funkciĭ Funkcional. Anal. i Priložen. Vyp. 1 1965 194–204. 

where this is stated as a Theorem but without proof.

I am looking for a reference containing a proof of this result.

Answer 1

You can find this result in the book of T. Kato. Perturbation theory for linear operators. Springer 1980, 1995. In Theorem IV.4.2, page 219.

Answer 2

References are boring if the proof is simple:

The open mapping theorem tells you that $L+M$ is closed if and only the canonical map $S:L\times M \to L+M$ $(x,y)\mapsto x-y$ is an isomorphism (where the product is endowed with the norm $\|(x,y)\|=\|x\|+\|y\|$). The inclination of $L\times\{0\}$ and $\{0\}\times M$ in this product is $1$ which gives you the necessity of $\widehat{(M,N)}>0$. For the other implication, positive inclination $c$ means $\|x-y\|\ge c\|x\|$ for all $(x,y)\in L\times M$ which gives you the continuity of the first component of $S^{-1}$, and this also implies the continuity of the second.

Answer 3

I think that in the meantime, I found the paper where the Theorem originally appeared. It seems to be

Grinblyum, M. M.: On the representation of a space of type B in the form of a direct sum of subspaces. Doklady Akad. Nauk SSSR 70, (1950). 749–752.

In this article the mentioned theorem is Theorem 1 on p. 749.