Let $(X, \|\cdot\|_X), (Y, \|\cdot\|_Y)$ normed spaces, operator $A: X \longrightarrow Y$ weakly continuous. Is $A$ continuous with respect to the strong topology?
Thanks in advance for your ideas.
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1A linear operator between normed spaces is weak-to-weak continuous if and only if it is norm-to-norm continuous, and each of these equivalent conditions is implied by weak-to-norm continuity and implies norm-to-weak continuity. I cannot presently think of any examples of weak-to-norm continuous operators which are not norm-to-norm continuous, but probably there are some, even for $X$ and $Y$ Banach. – Ben W Jan 26 '16 at 02:36
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1@anonymous, as far as I know all weak-to-norm continuous operators are finite rank – Norbert Jan 26 '16 at 07:09