I'm trying to understand what's wrong with my "proof" that $U = (U^\perp)^\perp$. I'm reading a book on finite-dimensional linear algebra (axler), and his proof is much more involved, and I know my proof must be wrong since my proof generalizes to infinite dimensions.

So...

(A) Show $U \subset (U^\perp)^\perp$. Let $v \in U$. Then v is orthogonal to every element in $U^\perp$, so v is also an element of $(U^\perp)^\perp$.

(B) $(U^\perp)^\perp \subset U$. So let $v \in (U^\perp)^\perp$. So v is orthogonal to every vector in $(U^\perp)$. By a previous theorem, we know that our vector space V is the direct sum of U and $U^\perp$, which means there are three possibilities.

1. v \in U
2. v \in $U^\perp$
3. or v = linear combination of vectors in U and $U^\perp$.

Since v is orthogonal to every vector in $U^\perp$, v cannot be in $U^perp$, as otherwise we would have <v,v> = 0, so this is only possible if v is the zero vector, so we exclude (2). Similar reasoning excludes (3), so we are left with (1).

Not sure where the hole is... any thoughts?

Thanks.