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The question reads like this:

Let $X$ be a reflexive Banach space and $K \subset X$ a set.

i) Given $r > 0$ define the application $T_r: X \rightarrow X$ as $T(x) = rx$. Show that $T_r$ is continuous considering in $X$ the weak topology in the domain and the counter-domain.

ii) Show that if $K$ is convex, bounded and closed in the strong topology, then $K$ is compact on the weak topology.

For the item i) i used that a linear application between Banach spaces is continuous when both spaces use the weak topology if and only if it's continuous when both use the strong topology. Since $T_r$ is bounded on the strong topology, it is continuous on it, and thus continuous on the weak topology.

I don't get how this should help with the item ii).

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    $\begingroup$ It helps maybe a little. The main thing to consider for ii) is Banach-Alaoglu. $\endgroup$ Aug 21, 2020 at 18:36

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Continuous functions map compact sets to compact sets. So i), together with Banach-Alaoglu, shows that all balls centered at $0$ are compact.

Being bounded, $K$ is inside a ball. Being convex and closed, it is weakly closed. A closed subset of a compact set, in a Hausdorff space, is compact.

Also, you don't really need any theory for i). If $x_n\to x$ weakly, this means that $f(x_n)\to f(x)$ for all $f\in X^*$. Now $$ f(T_rx_n)=f(rx_n)=rf(x_n)\to rf(x)=f(rx)=f(T_rx),\qquad f\in X^*. $$ So $T_rx_n\to T_rx$ weakly and $T_r$ is continuous (note that one is working with nets here, but nothing changes in the argument).

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