Knowlpedia

Linear Closure

The algebraic analogue of closure for subsets of vector spaces

Let XX be a real and let Ω⊆X\Omega\subseteq 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 the half-open is

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

When XX is a and Ω\Omega is ,

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

where Ω‾\overline{\Omega} is the usual .

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