Theorems · Definition · functional analysis

ContinuousLinearMap.extend

{𝕜 : Type u_1} →
  {𝕜₂ : Type u_2} →
    {E : Type u_3} →
      {Eₗ : Type u_4} →
        {F : Type u_5} →
          [inst : AddCommGroup E] →
            [inst_1 : UniformSpace E] →
              [IsUniformAddGroup E] →
                [inst_3 : AddCommGroup F] →
                  [inst_4 : UniformSpace F] →
                    [IsUniformAddGroup F] →
                      [T0Space F] →
                        [inst_7 : AddCommMonoid Eₗ] →
                          [inst_8 : UniformSpace Eₗ] →
                            [ContinuousAdd Eₗ] →
                              [inst_10 : Semiring 𝕜] →
                                [inst_11 : Semiring 𝕜₂] →
                                  [inst_12 : Module 𝕜 E] →
                                    [inst_13 : Module 𝕜₂ F] →
                                      [inst_14 : Module 𝕜 Eₗ] →
                                        [ContinuousConstSMul 𝕜 Eₗ] →
                                          [ContinuousConstSMul 𝕜₂ F] →
                                            {σ₁₂ : 𝕜 →+* 𝕜₂} →
                                              (E →SL[σ₁₂] F) → [CompleteSpace F] → (E →L[𝕜] Eₗ) → Eₗ →SL[σ₁₂] F

Extension of a continuous linear map f : E →SL[σ₁₂] F, with E a normed space and F a complete normed space, along a uniform and dense embedding e : E →L[𝕜] Eₗ.

Defined in

Mathlib.Analysis.Normed.Operator.Extend

Cited by

5 results in Mathlib

Foundations

Depth 97 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

AddCommGroupUniformSpaceIsUniformAddGroupAddCommGroupUniformSpaceIsUniformAddGroupT0SpaceAddCommMonoidUniformSpaceContinuousAddSemiringSemiringModuleModuleModuleContinuousConstSMulContinuousConstSMulCompleteSpace