Let $X$ be a Banach space and let $K$ be a (strongly) compact subset of $X$.
First $K$ is also weakly-compact because the weak topology is weaker than the strong topology.
Both trace topologies on $K$ coming from the strong and the weak topologies of $X$ are Hausdorff separated.
Recall that if on a set we give two comparable Hausdorff separated topologies that make that set into a compact topological space then those topologies coincide (on that set).
Therefore is it correct to say that the strong and weak topologies coincide on $K$?