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 $f(x)$.
Best Answer
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.