I'm working through Ward Cheneys Analysis for Applications and I'm a bit stuck on this exercise from Section 2.2:
Prove that if $M=M^{\perp\perp}$ for every closed linear subspace $M$ in an inner product space $V$, then $V$ is complete.
To clarify, $V$ is assumed to be a vector space over $\mathbb{C}$. I've thought about trying to prove the contrapositive. Given a Cauchy sequence $\left( x_n \right)$, one possible closed subspace we might consider is
$$U=\{ u\in V : \lim_{n\rightarrow\infty} \langle u,x_n \rangle =0 \},$$
but I really don't know how to proceed.