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In mathematics, specifically in functional analysis and order theory, a **topological vector lattice** is a Hausdorff topological vector space (TVS) that has a partial order making it into vector lattice that is possesses a neighborhood base at the origin consisting of solid sets.^{[1]}
Ordered vector lattices have important applications in spectral theory.

If is a vector lattice then by **the vector lattice operations** we mean the following maps:

- the three maps to itself defined by , , , and
- the two maps from into defined by and.

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

If is a vector lattice and an ordered topological vector space that is a Fréchet space in which the positive cone is a normal cone, then the lattice operations are continuous.^{[1]}

If is a topological vector space (TVS) and an ordered vector space then is called **locally solid** if possesses a neighborhood base at the origin consisting of solid sets.^{[1]}
A **topological vector lattice** is a Hausdorff TVS that has a partial order making it into vector lattice that is locally solid.^{[1]}

Every topological vector lattice has a closed positive cone and is thus an ordered topological vector space.^{[1]}
Let denote the set of all bounded subsets of a topological vector lattice with positive cone and for any subset , let be the -saturated hull of .
Then the topological vector lattice's positive cone is a strict -cone,^{[1]} where is a **strict -cone** means that is a fundamental subfamily of that is, every is contained as a subset of some element of ).^{[2]}

If a topological vector lattice is order complete then every band is closed in .^{[1]}

The Lᵖ spaces () are Banach lattices under their canonical orderings. These spaces are order complete for .