Continuous Strong-Strong Implies Continuous Weak-Weak

Asked Nov 13 '14 at 11:49

Active Nov 13 '14 at 11:49

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Let $X$ and $Y$ be two Banach spaces and let $T$ be a linear map between $X$ and $Y$. Show that $T$ is continuous strong-strong if and only if $T$ is continuous weak-weak.

I can see that $T$ being continuous strong-strong implies that $T$ is continuous strong-weak. How to see weak-weak, please?
I have no idea on the other direction. Could anyone help me, please? Thank you!

asked Nov 13 '14 at 11:49

LaTeXFan

3,548

Hint: Assume it's weak-weak but not strong-strong. Then follow a proof by contradiction. – Axoren Nov 13 '14 at 11:52
@Axoren Could add more detail, please? – LaTeXFan Nov 13 '14 at 11:54
You have to use the adjoint of the operator$T$. – Neutral Element Nov 13 '14 at 17:09
For weak-weak implying strong-strong see http://math.stackexchange.com/questions/5795/weak-sequential-continuity-of-linear-operators – daw Nov 14 '14 at 13:59

Continuous Strong-Strong Implies Continuous Weak-Weak

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