Let $X$ be a reflexive [Banach space](/page/Banach%20Space) and let $Y$ be a Banach space. For $T \in \mathcal{L}(X,Y)$, the following are equivalent:
\begin{align*}
&\text{1. } T \in \mathcal{K}(X,Y),\\
&\text{2. whenever } x_k \rightharpoonup x \text{ in } X, \text{ we have } \|Tx_k-Tx\|_Y \to 0.
\end{align*}