Do equivalent norms preserve dual spaces? Suppose that $X^*$ is the dual space of a normed space $X$. If we renorm the space $X^*$ with a new norm equivalent to the first one, is this new normed space the dual of $X$ as well?
(I think it suffices to prove that a functional $f$ is continuous with a norm1 if and only if it is continuous with norm2 where norm1 and norm2 are two equivalent norms. This seems to be obvious!).
Thanks for the help.
 A: No. Since you don't change the vector space $X^*$ by renorming it, it remains the dual as the set of continuous linear functionals on $X$. But it is no longer the dual of $X$ as a normed vector space, since that means precisely $\|f\|=\sup_{\|x\|\leq 1} |f(x)|$, which is completely determined by the norm on $X$.
Along these lines, what is true is: if you put an equivalent norm on $X$, then $X$ has the same bounded (= continuous) linear functionals as before, so the vector space $X^*$ remains the same. And both induced norms on $X^*$ are equivalent as well. Maybe that's what you meant to ask.
To prove the statement of that last paragraph, assume first that $\|x\|_1\leq C\|x\|_2$ for some $C>0$. Then $\|x\|_1\leq 1$ whenever $\|Cx\|_2\leq 1$, whence for every linear functional on $X$
$$
C\|f\|_1=C\sup_{\|x\|_1\leq 1}|f(x)|\geq \sup_{\|Cx\|_2\leq 1}|f(Cx)|= \sup_{\|y\|_2\leq 1}|f(y)|=\|f\|_2.
$$
In particular, $f$ is $\|\cdot\|_2$ bounded whenever it is $\|\cdot\|_1$ bounded. The result follows by symmetry.
A: For the last statement, suppose $f$ is continuous with respect to $\lVert \cdot \rVert_1$, and that $k,K > 0$ are so that $k \lVert v \rVert_1 < \lVert v \rVert_2 < K\lVert v \rVert_1$ for all $v \in X$. Let $\varepsilon > 0$ be given, and choose $\delta_0$ so that $\lvert f(v) - f(w) \rvert < \varepsilon$ whenever $\lVert v - w \rVert_1 < \delta_0$. Put $\delta = \delta_0/k$. Then for $\lVert v - w \rVert_2 < \delta$ you have $\lVert v - w \rVert_1 < \delta/k = \delta_0$, so $\lvert f(v)-f(w) \rvert < \varepsilon$, so $f$ is continuous with respect to $\lVert \cdot \rVert_2$
Note that this does not use linearity of $f$ and is a general fact about a continuous maps on metric spaces.
