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Question: Let $K$ an infinite compact metric space. An example of a weakly convergent sequence in $C(k)$ that is not strongly convergent (with the norm of the sup).

I really appreciate if anyone can give a tip, I tried to find it but I can't yet.

I know what $(x_{n})_{n\geq1}$ weakly convergent to $x$ if $\forall f \in C(K)^{*}$, $f((x_{n})_{n\geq1})$ strongly convergent to $f(x)$.

emma
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  • mind elaborating on what you mean by weak and strong convergence? – PStheman Oct 26 '22 at 15:19
  • $x_n \to x$ strong convergence in () , is to converge on the norm ( norm of sup). $x_n \to x$ weak convergence if it continues to converge for all continuous linear functions of (). Remember that the weak topology, it's all the "open" ones that make the $f$ continuous. – emma Oct 26 '22 at 16:56
  • There's an example on the Wikipedia page about uniform convergence I believe – gist076923 Oct 26 '22 at 17:04
  • see https://math.stackexchange.com/questions/823958/weak-convergence-of-continuous-functions for an easy to interpret characterization of weak convergence. – daw Oct 26 '22 at 17:21

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You need a sequence of continuous functions that converge pointwise to a continuous function but not uniformly. This rules out $f_n(t) = t^n$.

One example is $$ f_n(x) = \chi_{[\frac1{n+1},\frac1n]} \sin(\frac{2\pi} x). $$ It converges pointwise to zero and is uniformly bounded hence weakly convergent. Of course it does not converge strongly to zero.

daw
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