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One may show that $A$ is dense in $B$ and $A$ closed. For instance, if $A \subset \mathbb{C}^n$ is an algebraic variety that contains $\mathbb{Z}^n$, it is all of $\mathbb{C}^n$ (of course, I am using the Zariski topology here).

One way of showing this density is to use the Hahn-Banach theorem (an example is the Muntz-Szasz theorem, cf. here, and there are other such applications such as the Stone-Weierstrass theorem that can be proved similarly). If every continuous linear functional vanishing on $B$ vanishes on $A$ and $A,B$ are linear subspaces of a Banach space, then $A$ is dense in $B$.

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Show that $A$ is dense in $B$ and $A$ closed. For instance, if $A \subset \mathbb{C}^n$ is an algebraic variety that contains $\mathbb{Z}^n$, it is all of $\mathbb{C}^n$ (of course, I am using the Zariski topology here).

One way of showing this density is to use the Hahn-Banach theorem (an example is the Muntz-Szasz theorem, cf. here, and there are other such applications such as the Stone-Weierstrass theorem that can be proved similarly). If every continuous linear functional vanishing on $B$ vanishes on $A$ and $A,B$ are linear subspaces of a Banach space, then $A$ is dense in $B$.