Let $X$ be a [Banach space](/page/Banach%20Space), let $Y$ be a [normed vector space](/page/Normed%20Vector%20Space), and let $T: X \to Y$ be compact. If $x_k \rightharpoonup x$ weakly in $X$, then
\begin{align*}
\|Tx_k - Tx\|_Y &\to 0.
\end{align*}
Analysis
Discussion
This theorem says that compact operators turn weakly convergent sequences into norm-convergent image sequences. It is a central compactness principle in Banach space theory and PDE applications.
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