Let $ V $ be an inner product space.
Which of the following 2 statements are correct?
1.For each 2 subspaces $ S,T\subset V $ if follows that if $ S^{\perp}=T^{\perp} $ then $ S=T $ - I think that this is the correct statement, but according to the lecturer who solve the test that I took this question from, this is the false statement.
The second statement:
2.For each subspace of $ V $ exists an orthonormal bases (the vector in the bases are orthogonal to eachother, and each vector $ v $ follows that $ ||v||=1 $
Now I think that 2 is incorrect, because what about the zero subspace? does it have an orthonormal base?
I'll show also my "proof" for the first statement. I'll "prove" that $ S=T $ :
fix $ s\in S $ then for each vector $ v \in T^{\perp} $ it follows that $ <s,v>=0 $ because $ S^{\perp}=T^{\perp} $.
Thus, $ s\in\left(T^{\perp}\right)^{\perp}=T $ so $ S\subseteq T $.
In the same way we show that $ T\subseteq S $ and thus $ S=T $.
What's wrong with my proof?
Thanks in advance.