# Equivalent norms and density/separability

$V \subset H$ 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.

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?
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 $Y=V$.