TITLE: Hilbert space with 2 subspaces
QUESTION [0 upvotes]: In my homework I am trying to understand the concept of Hilbert spaces with the following simple problem:
Let H be a Hilbert space and A and B be closed subspaces of H so that $A^{\perp} = B^{\perp}$
Determine if A=B.
My idea is no because A and B could be disjoint and still meet the condition. However I am not sure and any input would be appreciated

REPLY [2 votes]: Since $B$ is a closed subspace, we have a decomposition $H=B \oplus B^{\perp}$. We will show that $A\subset B$. Let $a\in A$, $a=b_1+b_2$ with $b_1\in B,b_2\in\ B^{\perp}$. Now because $A^{\perp}=B^{\perp}$, projection  of $a$ onto $B^{\perp}$ is the same as projection of a onto $A^{\perp}$, which is $0$, since $a\in A$. Therefore $b_2=0$ and thus $a=b_1\in B$. By symmetry $B\subset A$.
{"set_name": "stack_exchange", "score": 0, "question_id": 3382674}