Orthogonal subspace of an orthogonal subspace

Question

Let $V$ be an inner product space over $\mathbb{F}(\mathbb{C}\ or\ \mathbb{R})$, and let $W$ be a subspace of $V$. Assuming $V$ is finite-dimensional, I have proved that $(W^{\perp})^{\perp} = W$ using some dimension considerations.

Is this also true for an infinite-dimensional space?

Answer 1

As Prahlad Vaidyanathan stated, this is, in general, not true. However, it is true that $W\subseteq (W^\perp)^\perp$.

Answer 2

In general you have $W^{\bot \bot} = \overline{\operatorname{sp} W}$.

For a simple example, take $l_2$ and let $W =\operatorname{sp} \{e_k \}$, a proper dense subspace of $l_2$. Then $W^\bot = \{0\}$ and so $W^{\bot \bot} = l_2$.