Are there some non-complete inner product spaces in which the equality holds:$M^{\perp\perp}=\overline{\operatorname{span} M}?$ Let $X$ be a Hilbert space and  $M\subset X$. We know that the following is true:
$$(M^{\perp})^{\perp}=\overline{\operatorname{span} M}.$$
But  I want to know is it true if $X$ is an inner product space .
Can someone help me prove it or give a counterexample? Thank you in advance.

I have tried some examples such as $X=C[0,1] $ and $M=\{f\in C[0,1]:f \text{ is constant}\}$,but all of them show the positive answer.
This identity holds in all inner product spaces. The usual proof uses Hahn-Banach and no completeness is needed.
This identity holds in all inner product spaces. The usual proof uses Hahn-Banach and no completeness is needed.