Theorems · Definition · functional analysis

ContinuousLinearMap.equivProdOfSurjectiveOfIsCompl

{𝕜 : Type u_1} →
  {E : Type u_2} →
    {F : Type u_3} →
      {G : Type u_4} →
        [inst : NontriviallyNormedField 𝕜] →
          [inst_1 : NormedAddCommGroup E] →
            [inst_2 : NormedSpace 𝕜 E] →
              [inst_3 : NormedAddCommGroup F] →
                [inst_4 : NormedSpace 𝕜 F] →
                  [inst_5 : NormedAddCommGroup G] →
                    [inst_6 : NormedSpace 𝕜 G] →
                      [CompleteSpace E] →
                        [CompleteSpace (F × G)] →
                          (f : E →L[𝕜] F) →
                            (g : E →L[𝕜] G) →
                              (↑f).range = ⊤ → (↑g).range = ⊤ → IsCompl (↑f).ker (↑g).ker → E ≃L[𝕜] F × G

If f : E →L[R] F and g : E →L[R] G are two surjective linear maps and their kernels are complement of each other, then x ↦ (f x, g x) defines a linear equivalence E ≃L[R] F × G.

Defined in

Mathlib.Analysis.Normed.Module.Complemented

Cited by

9 results in Mathlib

Foundations

Depth 172 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

NontriviallyNormedFieldNormedAddCommGroupNormedSpaceNormedAddCommGroupNormedSpaceNormedAddCommGroupNormedSpaceCompleteSpaceCompleteSpace