If $E$ is a Banach space, $A$ is a subset such that $$A^{\perp}:= \{T \in E^{\ast}: T(A)=0\}=0,$$ then $$\overline{A} = E.$$ I don't why this is true. Does $E$ has to be Banach? Thanks
Dense subset of given space
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functional-analysis
banach-spaces
normed-spaces