Linear Closure

The algebraic analogue of closure for subsets of vector spaces

Linear Closure

Let $X$ be a real vector space and let $\Omega\subset X$ .

The linear closure of $\Omega$ is

\operatorname{lin}(\Omega):=\Big\{x\in X\ \Big|\ \exists w\in\Omega \text{ with } [w,x)\subset \Omega\Big\},

where $[w,x)$ denotes the half-open line segment

[w,x):=\{\lambda w+(1-\lambda)x\mid \lambda\in(0,1]\}.

Equivalently, $x\in\operatorname{lin}(\Omega)$ iff there exists a point $w\in\Omega$ such that the entire segment from $w$ to $x$ (excluding $x$ ) lies in $\Omega$ .

When $X$ is a normed vector space and $\Omega$ is convex , we have

\Omega \subset \operatorname{lin}(\Omega)\subset \overline{\Omega},

where $\overline{\Omega}$ is the usual closure . See also algebraic interior (core) for the dual notion.

Examples:

If $\Omega$ is a linear subspace $L$ , then $\operatorname{lin}(L)=L$ .