Sentence examples for be a vector lattice from inspiring English sources

Definition 9 Let (E, ≤) be a vector lattice.

Theorem 31 Let (E, ≤) be a vector lattice and K be a regular cone.

Definition 18[5]Let (E, ≤) be a vector lattice and K be its positive cone.

Theorem 27 Let (E, ≤) be a vector lattice and K a regular cone in E. Let f : K → K be an order continuous and weakly increasing mappings with respect to the partial order induced by K and f ( x ) - f ( y ) ≤ ϕ ( x - y ) (2.3).

Theorem 25 Let (E, ≤) be a vector lattice and K a regular cone in E. Suppose that f, g : K → K are two mutually dominated mappings with respect to the order induced by K and f ( x ) - g ( y ) ≤ ϕ ( x - y ) (2.1).

A Banach lattice is a Banach space V endowed with an ordering ≤, such that ((V,leq)) is a vector lattice with a lattice norm defined on it.

A normed lattice X is a vector lattice with a norm ∥ ⋅ ∥ such that the following condition is satisfied: | x | ⪯ | y | implies ∥ x ∥ ≤ ∥ y ∥, for all  x, y ∈ X, where | x | is defined by | x | = x ∨ ( − x ) for each x ∈ X.

If furthermore the lattice property holds, that is, if for then is called a vector lattice.

An ordered vector space V is called a vector lattice, if any two elements (f,gin V) have a supremum, which is denoted by (sup f,g)) and an infimum denoted by (inf f,g)).

For f = g in Theorem 24, and in Theorem 25 we obtain the following results: Theorem 26 Let (E, ≤) be a complete vector lattice.

We establish the existence of common fixed points for weakly order contractive mappings by using only order-theoretic properties as follows: Theorem 24 Let (E, ≤) be a complete vector lattice and K its positive cone.