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In mathematics, specifically infunctional analysis andorder theory, atopological vector lattice is aHausdorfftopological vector space (TVS)${\displaystyle X}$ that has apartial order${\displaystyle \le }$ making it intovector lattice that possesses a neighborhood base at the origin consisting ofsolid sets.^[1] Ordered vector lattices have important applications inspectral theory.

If${\displaystyle X}$ is a vector lattice then bythe vector lattice operations we mean the following maps:

1. the three maps${\displaystyle X}$ to itself defined by${\displaystyle x\mapsto |x|}$,${\displaystyle x\mapsto x^{+}}$,${\displaystyle x\mapsto x^{-}}$, and
2. the two maps from${\displaystyle X\times X}$ into${\displaystyle X}$ defined by${\displaystyle (x,y)\mapsto \underset{}{sup}\{x,y\}}$ and${\displaystyle (x,y)\mapsto \underset{}{inf}\{x,y\}}$.

If${\displaystyle X}$ is a TVS over the reals and a vector lattice, then${\displaystyle X}$ is locally solid if and only if (1) its positive cone is anormal cone, and (2) the vector lattice operations are continuous.^[1]

If${\displaystyle X}$ is a vector lattice and anordered topological vector space that is aFréchet space in which the positive cone is anormal cone, then the lattice operations are continuous.^[1]

If${\displaystyle X}$ is atopological vector space (TVS) and anordered vector space then${\displaystyle X}$ is calledlocally solid if${\displaystyle X}$ possesses a neighborhood base at the origin consisting ofsolid sets.^[1] Atopological vector lattice is aHausdorff TVS${\displaystyle X}$ that has apartial order${\displaystyle \le }$ making it intovector lattice that is locally solid.^[1]

Every topological vector lattice has a closed positive cone and is thus anordered topological vector space.^[1] Let${\displaystyle \mathcal{B}}$ denote the set of all bounded subsets of a topological vector lattice with positive cone${\displaystyle C}$ and for any subset${\displaystyle S}$, let${\displaystyle [S{]}_C:=(S+C)\cap (S-C)}$ be the${\displaystyle C}$-saturated hull of${\displaystyle S}$. Then the topological vector lattice's positive cone${\displaystyle C}$ is a strict${\displaystyle \mathcal{B}}$-cone,^[1] where${\displaystyle C}$ is astrict${\displaystyle \mathcal{B}}$-cone means that${\displaystyle \left\{[B{]}_C:B\in \mathcal{B}\right\}}$ is a fundamental subfamily of${\displaystyle \mathcal{B}}$ that is, every${\displaystyle B\in \mathcal{B}}$ is contained as a subset of some element of${\displaystyle \left\{[B{]}_C:B\in \mathcal{B}\right\}}$).^[2]

If a topological vector lattice${\displaystyle X}$ isorder complete then every band is closed in${\displaystyle X}$.^[1]

TheL^p spaces (${\displaystyle 1\le p\le \mathrm{\infty }}$) areBanach lattices under their canonical orderings. These spaces are order complete for.

• Banach lattice – Banach space with a compatible structure of a lattice
• Complemented lattice
• Fréchet lattice – Topological vector lattice
• Locally convex vector lattice
• Normed lattice
• Ordered vector space – Vector space with a partial order
• Pseudocomplement
• Riesz space – Partially ordered vector space, ordered as a lattice

• Narici, Lawrence; Beckenstein, Edward (2011).Topological Vector Spaces. Pure and applied mathematics (Second ed.). Boca Raton, FL: CRC Press.ISBN 978-1584888666.OCLC 144216834.
• Schaefer, Helmut H.; Wolff, Manfred P. (1999).Topological Vector Spaces.GTM. Vol. 8 (Second ed.). New York, NY: Springer New York Imprint Springer.ISBN 978-1-4612-7155-0.OCLC 840278135.

• Functional analysis

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