= Compact closed-ball image forces finite rank on c0
{title2=$S(B_{c_0})\text{ compact}\Longrightarrow\operatorname{rank}S<\infty$}
Suppose a linear map $S:c_0\to Y$ has compact <image> of the closed unit ball. It is bounded, and each $g\in Y^*$ attains its maximum on this compact <image>. Thus $S^*g$ attains its <norm>, so <norm attainment on the vanishing sequence space> implies $S^*(Y^*)\subseteq c_{00}$. The <Baire category theorem> applied to the inverse images of the first $n$ coordinate spans makes this adjoint finite rank: these are closed <linear subspaces> covering the complete <continuous dual space> $Y^*$, so one has interior and is all of $Y^*$. Finally the <Hahn-Banach theorem> identifies $\ker S$ as the common kernel of $S^*(Y^*)$, proving finite rank of $S$. Ordinary <compact operators> require only relative compactness of the ball <image>, so they need not have finite rank.
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