Theorems · Definition · functional analysis

ContinuousLinearMap.coprodSubtypeLEquivOfIsCompl

{𝕜 : Type u_1} →
  [inst : NontriviallyNormedField 𝕜] →
    {E : Type u_3} →
      [inst_1 : NormedAddCommGroup E] →
        [inst_2 : NormedSpace 𝕜 E] →
          [CompleteSpace E] →
            {F : Type u_5} →
              [inst_4 : NormedAddCommGroup F] →
                [inst_5 : NormedSpace 𝕜 F] →
                  [CompleteSpace F] →
                    (f : E →L[𝕜] F) →
                      {G : Submodule 𝕜 F} → IsCompl (↑f).range G → [CompleteSpace ↥G] → (↑f).ker = ⊥ → (E × ↥G) ≃L[𝕜] F

Intermediate definition used to show ContinuousLinearMap.closed_complemented_range_of_isCompl_of_ker_eq_bot. This is f.coprod G.subtypeL as a ContinuousLinearEquiv.

Defined in

Mathlib.Analysis.Normed.Operator.Banach

Cited by

2 results in Mathlib

Foundations

Depth 173 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

NontriviallyNormedFieldNormedAddCommGroupNormedSpaceCompleteSpaceNormedAddCommGroupNormedSpaceCompleteSpaceCompleteSpace