1. The problem statement, all variables and given/known data 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. 2. Relevant equations I believe that (u,v)=0 is an inner product space 3. 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.  I came across some new info. It has to do with whether N is finite dimensional or not. If N is finite dimentional 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.