Algebra: Orthogonal Complement

Question

Problem

Let $V$ be a real inner product space and $U \subset V$. Show that $(U^\perp)^\perp=U$.

Progress

Clearly for $x\in U$ we have that $\langle x,v \rangle=0$ for all $v \in U^\perp$. This immediately yields that $x \in (U^\perp)^\perp$ and so $U \subset (U^\perp)^\perp$.

Taking $x \in (U^\perp)^\perp$, we have that $\langle x,v \rangle=0$ for all $v \in U^\perp$. Not sure how to move it on from here though; any assistance would be much appreciated. Regards.

Answer 1

Since you already know that $U\subseteq U^{\perp\perp}$, it will suffice that these two spaces have the same dimension. Since $$\dim(U^{\perp\perp})=\dim(V)-\dim(U^\perp)=\dim(V)-\dim(V)+\dim(U)=\dim(U),$$ you are done.

Answer 2

If $\dim V = n$ and $\dim U = k$, then $\dim U^\perp = n-k$. Then argue using orthonormal basis.

Answer 3

Hint: use the orthogonal decomposition $V=U\oplus U^\perp$.