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I am reading Royden’s real analysis. In his book, a sequence of functions in Lp converges weakly if every bounded linear functional in the dual space converges in R.

Can anyone discuss the importance of weak convergence and give an example of a sequence that converges weakly, but not strongly?

user1559897
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  • For example, take an infinite-dimensional separable Hilbert space and an orthonormal basis $(\phi_k)$. Then $\phi_k$ converges weakly to zero, but not strongly (check the former using Riez). – Daniel Jun 16 '16 at 21:50
  • I am new this stuff and haven't got to the chapter on Hilbert space yet... – user1559897 Jun 16 '16 at 21:53
  • Do you know the sequence spaces $\mathcal{l}^p$? If yes, $l^2$ is a separable Hilbert space you can work with. If not, I would recommend reading a bit further and then coming back to this. – Daniel Jun 16 '16 at 21:58
  • Yep, I know about L2. Just not Hilbert space yet. Can give a concrete example of the basis and weak convergence? – user1559897 Jun 16 '16 at 22:03

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