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.
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$\begingroup$ Hmm I think taht gives an answear to a weaker question because I required that dual spaces are the same not isomorphic, so a given functional is continuous in one norm iff in the second one. $\endgroup$ – J.E.M.S Jun 17 '16 at 22:11
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$\begingroup$ You unnecesarry deleted your comment it was helpful in a way. $\endgroup$ – J.E.M.S Jun 18 '16 at 9:39
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Yes.
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.
answered Jun 17 '16 at 23:08
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$\begingroup$ For what it's worth, professor, you might use '\| \|' to render $\|\cdot\|$ which looks better than the $||\cdot||$ rendered from '|| ||'. $\endgroup$ – Vim Jun 18 '16 at 14:27
