I got this questions on one of my example sheets in a first year algebra course:
"Let $B:W×V→K$ be a bilinear map, where $V,W$ are vector spaces over a a field $K$. Let $U$ be a subspace of $V$. If we denote by $U^{\perp}= \{w \in W|B(w,u)=0,\forall u∈U \},$ prove that $dim(U)+dim(U^{\perp})≥dim(W)$.
I managed to come up with a proof in the case when $V,W$ are finite dimensional by considering a basis of $V$ which contains a basis of $U$, a basis of $W$ and then considering the coordinate matrix of $B$.
I can't think of an argument which works for the infinite dimensional case. Could somebody suggest me something? Thank you