# Sum of subspaces is closed iff inclination is positive

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.

• You should assume something else. Otherwise $M=N$ where $M$ is a closed subspace of $X$ is an obvious counterexample. – fedja Nov 21 at 14:24
• @fedja: thank you, I fixed it. – Christian Nov 21 at 14:27

## 3 Answers

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.

• The result in Kato's book is a bit more general. It does not need $M\cap N=\{0\}$. – M.González Nov 23 at 8:42

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.

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.