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Suppose we have two Hilbert spaces $H_1,~ H_2$, a linear continuous operator $T:H_1 \to H_2$ and a weakly convergent sequence $u_k\rightharpoonup u$ in $H_1$.

Is $Tu_k \rightharpoonup Tu$ in $H_2$ true? If yes, could you give me a short sketch of the proof?

I really appreciate any help you can provide.

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    $\begingroup$ Hint: T has an adjoint. $\endgroup$ – Nate Eldredge Sep 16 '15 at 13:56
  • $\begingroup$ You're right, thanks for the hint! $\endgroup$ – fmeyer Sep 16 '15 at 13:59
  • $\begingroup$ Your assertion is true even in Banach spaces. $\endgroup$ – julian Jan 10 '18 at 13:51

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