Definition:Compact Space/Normed Vector Space

Definition
Let $(X,\norm \cdot)$ be a normed vector space.

Let $K \in X$.

Then $K$ is compact if and only if every sequence in $K$ has a convergent subsequence with limit $L \in K$.

That is, if:


 * $(x_n)_{n \in \N} :\forall n \in \N : x_n \in K \implies \exists \paren{x_{n_k}}_{k \in \N} : \exists L \in K: \displaystyle \lim_{k \mathop \to \infty} x_{n_k} = L$.