Definition:Compact Space/Normed Vector Space/Subspace

Definition
Let $M = \struct{X, \norm {\,\cdot\,}}$ be a normed vector space.

Let $K \subseteq X$ be a subset of $X$.

The normed vector subspace $M_K = \struct {K, \norm {\,\cdot\,}_K}$ is compact in $M$ $M_K$ is itself a compact normed vector space.