It is trivial that if we have a vector space $X$ and two equivalent norms on it than $X'_1$ -dual space (of continuous functionals) for the first norm and $X'_2$ are the same spaces. Is the converse true? I asked myself and doubted at first glance, yet I failed to produce any counterexample. I will appriciate any kind of help.
The Hahn-Banach theorem shows that if $Y$ is a normed vector space and $y\in Y$ then $||y||_Y$ is equal to the norm of $y$ regarded as a linear functional on $Y'$.
Say $X_1$ and $X_2$ are normed spaces consisting of the same space with two different norms, and $X_1'=X_2'$. We need to show that if $||x_n||_1=1$ for $n=1,2\dots$ then $||x_n||_2$ is bounded. So we need to show that $(x_n)$ is a norm-bounded subset of $X_2''$. By the Uniform Boundedness Principle we need only show that $f(x_n)$ is bounded for every $f\in X_2'$. But this is clear because $f\in X_1'$ and $||x_n||_1$ is bounded.