Let $X$ and $Y$ be normed spaces, and let $T \in \mathcal{L}(X,Y)$. Then $T$ is compact if and only if for every bounded sequence $(x_k)_{k=1}^\infty$ in $X$, the sequence $(Tx_k)_{k=1}^\infty$ has a subsequence that converges in $Y$.
AnalysisFunctional Analysis
Discussion
Characterises compact operators between normed spaces by their action on bounded sequences: every bounded sequence in the domain has an image subsequence converging in the codomain.
Proof
No proof available for this theorem.
Prerequisites
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Prerequisites Graph
Interactive dependency map showing how this theorem builds on foundational concepts