Definition:Compact Space/Normed Vector Space

Definition

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

Let $K \subseteq X$.

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

That is, if:

$\sequence {x_n}_{n \mathop \in \N} :\forall n \in \N : x_n \in K \implies \exists \sequence {x_{n_k} }_{k \mathop \in \N} : \exists L \in K: \ds \lim_{k \mathop \to \infty} x_{n_k} = L$

Compact Subspace

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 $M$ if and only if $M_K$ is itself a compact normed vector space.

Also see

• Results about compact spaces can be found here.

Sources

• 2017: Amol Sasane: A Friendly Approach to Functional Analysis ... (previous) ... (next): Chapter $\S 1.5$: Normed and Banach spaces. Compact sets