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$V \subset H$ are Hilbert spaces with inner products $(\cdot,\cdot)_V$ and $(\cdot,\cdot)_H$. Suppose $V$ is dense in $H$ and both spaces are separable. If $(\cdot,\cdot)_{V_2}$ and $(\cdot,\cdot)_{H_2}$ are different inner products with norms equivalent to the original inner products, is $V \subset H$ with these new inner products/norms still dense and separable?

I say yes.

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Both questions are in principle the same as they can be reduced to the following two questions

  1. If I equip a separable normed space $X$ with an equivalent norm, is it still separable?
  2. If I equip a normed space $X$ with an equivalent norm, is then every dense subset still dense?

However, the first question can be reduced to the second one, since it only asks whether certain dense subsets (the countable ones) are still dense in the new norm.

Now it's easy to see that a set that is dense in one norm will also be dense in an equivalent norm. E.g. let $Y$ be such a set that is dense in $X$ with respect to the norm $\|\cdot\|_X$. Let $x\in X$ and $\epsilon>0$ and let $\|x\|_{X_2}\leq\alpha\|x\|_X$ for some $\alpha>0$ and all $x$ (because the norms are equivalent). Then you can find $y\in Y$ such that $\|y-x\|_{V}\leq\epsilon/\alpha$ and thus $\|y-x\|_{V_2}\leq\epsilon$.

When you ask for separability of $V$, set $X=V$ and $Y$ is any countable dense subset. When you ask whether $V$ is dense in $H$, set $X=H$ and $Y=V$.

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