\begin{definition}[Definition:Compact Space/Normed Vector Space]
Let $\struct {X, \norm {\,\cdot\,} }$ be a normed vector space. 
Let $K \subseteq X$.
Then $K$ is '''compact''' {{iff}} every sequence in $K$ has a convergent subsequence with limit $L \in K$.
That is, if:
:$\sequence {x_n}_{n \mathop \in \N} :\forall n \in \N : x_n \in K \implies \exists \sequence {x_{n_k} }_{k \mathop \in \N} : \exists L \in K: \ds  \lim_{k \mathop \to \infty} x_{n_k} = L$
\end{definition}