Let $(X,\|\cdot\|_X)$ be a [normed vector space](/page/Normed%20Vector%20Space), let $(Y,\|\cdot\|_Y)$ be a [Banach space](/page/Banach%20Space), and let $T:X\to Y$ be a [linear map](/page/Linear%20Map). Then $T$ is compact, meaning that $\overline{T(A)}^{\,Y}$ is compact in $Y$ for every bounded subset $A\subset X$, if and only if for every bounded sequence $(x_k)_{k\in\mathbb N}$ in $X$, the sequence $(Tx_k)_{k\in\mathbb N}$ has a subsequence converging in $Y$.