Locally convex vector lattice

In mathematics, specifically in order theory and functional analysis, a locally convex vector lattice (LCVL) is a topological vector lattice that is also a locally convex space.Template:Sfn LCVLs are important in the theory of topological vector lattices.

Lattice semi-norms

The Minkowski functional of a convex, absorbing, and solid set is a called a lattice semi-norm. Equivalently, it is a semi-norm $p$ such that $|y|\le |x|$ implies $p(y)\le p(x).$ The topology of a locally convex vector lattice is generated by the family of all continuous lattice semi-norms.Template:Sfn

Properties

Every locally convex vector lattice possesses a neighborhood base at the origin consisting of convex balanced solid absorbing sets.Template:Sfn

The strong dual of a locally convex vector lattice $X$ is an order complete locally convex vector lattice (under its canonical order) and it is a solid subspace of the order dual of $X$; moreover, if $X$ is a barreled space then the continuous dual space of $X$ is a band in the order dual of $X$ and the strong dual of $X$ is a complete locally convex TVS.Template:Sfn

If a locally convex vector lattice is barreled then its strong dual space is complete (this is not necessarily true if the space is merely a locally convex barreled space but not a locally convex vector lattice).Template:Sfn

If a locally convex vector lattice $X$ is semi-reflexive then it is order complete and $X_b$ (that is, $(X,b(X,X^{\prime }))$) is a complete TVS; moreover, if in addition every positive linear functional on $X$ is continuous then $X$ is of $X$ is of minimal type, the order topology ${\tau }_{\mathrm{O}}$ on $X$ is equal to the Mackey topology $\tau (X,X^{\prime }),$ and $(X,{\tau }_{\mathrm{O}})$ is reflexive.Template:Sfn Every reflexive locally convex vector lattice is order complete and a complete locally convex TVS whose strong dual is a barreled reflexive locally convex TVS that can be identified under the canonical evaluation map with the strong bidual (that is, the strong dual of the strong dual).Template:Sfn

If a locally convex vector lattice $X$ is an infrabarreled TVS then it can be identified under the evaluation map with a topological vector sublattice of its strong bidual, which is an order complete locally convex vector lattice under its canonical order.Template:Sfn

If $X$ is a separable metrizable locally convex ordered topological vector space whose positive cone $C$ is a complete and total subset of $X,$ then the set of quasi-interior points of $C$ is dense in $C.$Template:Sfn

If $(X,\tau )$ is a locally convex vector lattice that is bornological and sequentially complete, then there exists a family of compact spaces ${\left(X_{\alpha }\right)}_{\alpha \in A}$ and a family of $A$-indexed vector lattice embeddings $f_{\alpha }:C_{ℝ}\left(K_{\alpha }\right)\to X$ such that $\tau$ is the finest locally convex topology on $X$ making each $f_{\alpha }$ continuous.Template:Sfn

Examples

Every Banach lattice, normed lattice, and Fréchet lattice is a locally convex vector lattice.

See also

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Bibliography

• Template:Narici Beckenstein Topological Vector Spaces
• Template:Schaefer Wolff Topological Vector Spaces