Uniform space structure on matrices

instance Matrix.instUniformSpaceMatrix (m : Type u_1) (n : Type u_2) (𝕜 : Type u_3) [UniformSpace 𝕜] : UniformSpace (Matrix m n 𝕜)

Equations

• Matrix.instUniformSpaceMatrix m n 𝕜 = inferInstance

instance Matrix.instUniformAddGroup (m : Type u_1) (n : Type u_2) (𝕜 : Type u_3) [UniformSpace 𝕜] [AddGroup 𝕜] [UniformAddGroup 𝕜] : UniformAddGroup (Matrix m n 𝕜)

Equations

• (_ : UniformAddGroup (Matrix m n 𝕜)) = (_ : UniformAddGroup (m → n → 𝕜))

theorem Matrix.uniformity (m : Type u_1) (n : Type u_2) (𝕜 : Type u_3) [UniformSpace 𝕜] : uniformity (Matrix m n 𝕜) = ⨅ (i : m), ⨅ (j : n), Filter.comap (fun (a : Matrix m n 𝕜 × Matrix m n 𝕜) => (a.fst i j, a.snd i j)) (uniformity 𝕜)

theorem Matrix.uniformContinuous (m : Type u_1) (n : Type u_2) (𝕜 : Type u_3) [UniformSpace 𝕜] {β : Type u_4} [UniformSpace β] {f : β → Matrix m n 𝕜} : UniformContinuous f ↔ ∀ (i : m) (j : n), UniformContinuous fun (x : β) => f x i j

instance Matrix.instCompleteSpaceMatrixInstUniformSpaceMatrix (m : Type u_1) (n : Type u_2) (𝕜 : Type u_3) [UniformSpace 𝕜] [CompleteSpace 𝕜] : CompleteSpace (Matrix m n 𝕜)

Equations

• (_ : CompleteSpace (Matrix m n 𝕜)) = (_ : CompleteSpace (m → n → 𝕜))

instance Matrix.instSeparatedSpaceMatrixInstUniformSpaceMatrix (m : Type u_1) (n : Type u_2) (𝕜 : Type u_3) [UniformSpace 𝕜] [SeparatedSpace 𝕜] : SeparatedSpace (Matrix m n 𝕜)

Equations

• (_ : SeparatedSpace (Matrix m n 𝕜)) = (_ : SeparatedSpace (m → n → 𝕜))