I recently faced a problem concerning linear operators on Hilbert spaces, and I proved a small result. It finally appeared that I didn’t need it, but I have decided to record it anyway. Since it is very simple, I have absolutely no doubt that it has already been proved, but in keeping with the spirit of this blog, I don’t really care. I will keep it short, so there will be no discussion or motivation.
Let and be two Hilbert spaces. Let be a linear bounded operator. If is a closed subspace of and if is finite dimensional, the image by of any closed subspace of is a closed subspace of .
The proof I found relies on the following Lemma.
If is a normed linear space, a closed subspace of and a finite dimensional subspace of , then is a closed subspace of .
Let us first note that we only have to deal with the case where Indeed, in the general case, we can choose a complementary vector space in relative to . The space is obviously finite dimensional and .
Let us assume from now on that We denote by the unit ball of . It is a compact set for the topology given by the original norm on . The application (for the distance deduced from the norm) is continuous and positive on . Therefore, there exists such that for all . By homogeneity, this implies that for all .
Now let be a sequence of vectors in that converges to . There exist and , sequences of vectors in and respectively, such that . Now, according to the previous paragraph,
The sequence is therefore bounded. Since is finite dimensional, this implies the existence of and a subsequence such that . And now we are finished: , and therefore, being closed, . We obtain .
We can now prove the proposition. Let us consider , the restriction of . Since and are closed subspaces of and respectively (this is not automatic for , we had to make the hypothesis), they are both Hilbert spaces. Furthermore, is continuous, and bijective by construction. According to Banach’s Closed Graph Theorem, it has a bounded inverse . This implies, among other things, that the image of closed subsets of are closed subsets of , and thus of .
Let us now consider , a closed subspace of . It is easy to check that . According to the lemma, is closed and thus is a closed subspace of . According to what we have just seen, is closed, which conclude the proof.
I think that the proposition could be extended to Banach spaces. It should be enough to replace the orthogonal complement by the quotient space, but I need to revise some Functional Analysis before writing it up. The lemma leads to an interesting question: when is the sum of two closed vector spaces a closed vector space. A mathoverflow discussion deals with this topic, but here again I need to think about it a little longer before writing something.