[see also: change, place, instead]
We now construct $f$ as in the proof of Theorem 5, with $V$ replaced by $W$.
A similar result holds with `compact' replacing `convex'.
The statement of Theorem 5 remains valid if we replace `$f$ is compact' by `the norm of $f$ is bounded'.
We must have $Lf=0$, for otherwise we can replace $f$ by $f-Lf$.
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