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.



Sum of subspaces is closed iff inclination is positive
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. 


