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Let $(X,\|\cdot\|)$ be a Banach space and $(X^*,\|\cdot\|_{op})$ its continuous dual Banach space.

Let $\text{End}_{op}(X^*)$ be the continuous linear endomorphisms of $X^*$ with respect to $\|\cdot\|_{op}$ and $\text{End}_*(X^*)$ the continuous linear endomorphisms with respect to the weak-* topology of $X^*$.

Does $\text{End}_{op}(X^*)=\text{End}_*(X^*)$? If not when are they equal and how do they compare in general? Also references and proofs are welcome.


marked as duplicate by Norbert functional-analysis Oct 21 '17 at 10:13

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  • $\begingroup$ I don't think I've ever seen anything about operators on TVS, actually. Could you tell, what inspired this question? $\endgroup$ – Petr Naryshkin Oct 20 '17 at 22:17