Knowlpedia

Linear Closure

The algebraic analogue of closure for subsets of vector spaces

Let XX be a real and let Ω⊂X\Omega\subset X.

The linear closure of Ω\Omega is

lin⁡(Ω):={x∈X ∣ ∃w∈Ω with [w,x)⊂Ω},\operatorname{lin}(\Omega):=\Big\{x\in X\ \Big|\ \exists w\in\Omega \text{ with } [w,x)\subset \Omega\Big\},

where [w,x)[w,x) denotes the half-open

[w,x):={λw+(1−λ)x∣λ∈(0,1]}.[w,x):=\{\lambda w+(1-\lambda)x\mid \lambda\in(0,1]\}.
Equivalent characterizations

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

Remarks

When XX is a and Ω\Omega is , we have

Ω⊂lin⁡(Ω)⊂Ω‾,\Omega \subset \operatorname{lin}(\Omega)\subset \overline{\Omega},

where Ω‾\overline{\Omega} is the usual . See also for the dual notion.

Examples
  • If Ω\Omega is a linear subspace LL, then lin⁡(L)=L\operatorname{lin}(L)=L.