Let $F$ and $H$ be normed spaces and let $E$ be a locally convex space.
Let $T:F\to H$ and $S:H\to E$ be linear operators, such that $\|T\|= 1$, $S$ is an injective semi-embedding (i.e. $S\overline{B}\_{H}$ is closed in $E$), and $ST$ is weakly compact.



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> Does it follow that $T$ is weakly compact?
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The intuition is that if we consider $F^{\*\*}\to ^{T^{\*\*}} H^{\*\*}\to^{S^{\*\*}} E^{\*\*}$, then $S^{\*\*}T^{\*\*}\overline{B}\_{F^{\*\*}}$ is equal to the closure of $ST\overline{B}\_{F}$ in $E$, due to weak compactness of $ST$. Since $\|T\|=1$, it follows that $ST\overline{B}\_{F}\subset S\overline{B}\_{H}$, which is closed in $E$, and so $S^{\*\*}T^{\*\*}\overline{B}\_{F^{\*\*}}\subset S\overline{B}\_{H}$. Hence, perhaps it is possible to show that $T^{\*\*}\overline{B}\_{F^{\*\*}}\subset \overline{B}\_{H}$, which is equivalent to weak compactness of $T$.



Semi-embeddings and weak compactness
No. Let $S$ be the formal inclusion from $L\_\infty(0,1)$ into $L\_1(0,1)$ and let $T$ be the identity on $L\_\infty(0,1)$.


