- #1

Fellowroot

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## Homework Statement

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Let M, N be a subset of a Hilbert space and be two closed linear subspaces. Assume that (u,v)=0, for all u in M and v in N. Prove that M+N is closed.

## Homework Equations

I believe that (u,v)=0 is an inner product space

## The Attempt at a Solution

This is a problem from Haim Brezis's functional analysis book. It seems to be closely related to a typical linear algebra problem but only with Hilbert spaces.

The best thing I could find on this was this, but I need a little help showing this.

[edit]

I came across some new info. It has to do with whether N is finite dimensional or not. If N is finite dimensional then yes, M+N can be closed, but it may not be closed if N is infinite dimensional. Apparently I'm supposed to show this with induction on the dimension of N. Can anyone help on this part.

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