Given a bilinear form $f(a,b)$, we call it degenerate if for some $a$, $f(a,-)=0$. In the finite dimensional case, we have a matrix representation and can deduce from matrix theory that if $f$ is degenerate, there is some $b$ such that $f(-,b)=0$. But in the infinite dimensional case, it does not always apply.
Question is, does the equality hold if the vector space is infinite dimensional, or is there a counterexample?