Let $A \subset X$.
When do we have $A^\bot=\{0\}\implies\overline{A}=X \ \ \ \ ?$
I've read some answers in the forum that stated that for the converse $A$ had to be a subspace of $X$.
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https://math.stackexchange.com/questions/831897/trivial-orthogonal-complement-implies-denseness – An old man in the sea. Dec 10 '19 at 00:15
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A trivial situation: $A=[-1,1]\subseteq X=\mathbb R$. – Jochen Dec 10 '19 at 08:04