Sentence examples for be a generalized uniform from inspiring English sources

(c) Let (X, ) be a generalized uniform space.

Remark 2 Let (X, ) be a generalized uniform space.

Definition 5 Let (X, ) be a generalized uniform space.

(e) Let (X, ) be a generalized uniform space and let (x m : m ∈ ℕ) be a sequence in X.

Definition 6 Let (X, ) be a generalized uniform space, let L ∈ L ( X, D ) and let (x m : m ∈ ℕ) be a sequence in X.

Remark 3 Let (X, ) be a generalized uniform space and let L ∈ L ( X, D ). (i) If ∀ α ∈ A ∀ x ∈ X L α ( x, x ) = 0, then is a -family on X; examples of L ∈ L ( X, D ) which are not -families on X are given in Section "Examples of the decompositions of the generalized uniform spaces".

Indeed, if X is a vector space over ℝ and (X, ) is a generalized locally convex space, then D = d α : X × X → [ 0, + ∞ ], α ∈ A where d α (x,y) = p α (x - y), (x,y) ∈ X × X, α ∈ A, is -family and (X, ) is a generalized uniform space.

Remark 6 It is clear that each generalized locally convex space is an generalized uniform space.

(b) If is -family, then the pair (X, ) is called a generalized uniform space.

Definition 8 Let (X, ) be a Hausdorff generalized uniform space and let L ∈ L ( X, D ).

Example 5 Let (X, ) be a Hausdorff generalized uniform space where D = d α : X × X → [ 0, + ∞ ], α ∈ A, -index set, is a -family.