If $x_n\to x$ in a normed space, then $\|x_n\|\to \|x\|$, and $x_n\to x$ weakly. If the space is finite-dimensional, or it is a Hilbert space, the converse is also true: weak convergence+convergence of the norms implies convergence in norm. Are there any other non-trivial examples when this converse implication holds? Or any counterexamples?
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$\ell^1$ is another example where weak convergence implies that in norm (look up Schur's theorem). – Chrystomath Feb 11 '21 at 15:41
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https://math.stackexchange.com/questions/163209/weak-convergence-in-lp-plus-convergence-of-norm-implies-strong-convergence – daw Feb 11 '21 at 15:49
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This identity is often called "Radon-riesz (or kadets-klee)" property. It is known that every uniformly convex (also known as uniformly rotund) normed space has the Radon-reisz property. See for example "An Introduction to Banach Space Theory, by Robert Megginson" theorem 5.2.18 – vectorSpace Feb 11 '21 at 17:30
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Thanks for all the replies, they answer my question perfectly. Uniform convexity is fascinating :-) – user60121 Feb 12 '21 at 07:16
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@vectorSpace That qualifies as answer, you can write it down below and ask user60121 to accept. – Sarvesh Ravichandran Iyer Feb 13 '21 at 11:58