# Sequential compactness

The Bolzano-Weierstrass theorem states that every bounded sequence has a convergent subsequence.

Sequential compactness extends this notion to general topological spaces.

## Definition

A topological space [ilmath](X,\mathcal{J})[/ilmath] is sequentially compact if every (infinite) Sequence has a convergent subsequence.

### Common forms

#### Functional Analysis

A subset [ilmath]S[/ilmath] of a normed vector space [math](V,\|\cdot\|,F)[/math] is sequentially compact if any sequence [math](a_n)^\infty_{n=1}\subset k[/math] has a convergent subsequence [math](a_{n_i})_{i=1}^\infty[/math], that is [math](a_{n_i})_{i=1}^\infty\rightarrow a\in K[/math]

Like with compactness, we consider the subspace topology on a subset, then see if that is compact to define "compact subsets" - we do the same here. As warned below a topological space is not sufficient for sequentially compact [math]\iff[/math] compact, so one ought to use a metric subspace instead. Recalling that a norm can give rise to the metric [math]d(x,y)=\|x-y\|[/math]

## Warning

Sequential compactness and compactness are not the same for a general topology

## Uses

- A metric space is compact if and only if it is sequentially compact, a theorem found here