Are there some non-complete inner product spaces in which the equality holds:$M^{\perp\perp}=\overline{\operatorname{span} M}?$

Question

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.

Answer 1

This identity holds in all inner product spaces. The usual proof uses Hahn-Banach and no completeness is needed.