Theorems · Definition · functional analysis

ContinuousLinearMap.ofIsTopCompl

{R : Type u_1} →
  [inst : Ring R] →
    {E : Type u_2} →
      {F : Type u_3} →
        [inst_1 : TopologicalSpace E] →
          [inst_2 : AddCommGroup E] →
            [inst_3 : Module R E] →
              [IsTopologicalAddGroup E] →
                [inst_5 : TopologicalSpace F] →
                  [inst_6 : AddCommGroup F] →
                    [inst_7 : Module R F] →
                      [ContinuousAdd F] →
                        {p q : Submodule R E} → Submodule.IsTopCompl p q → (↥p →L[R] F) → (↥q →L[R] F) → E →L[R] F

Given continuous linear maps φ : p →L[R] F and ψ : q →L[R] F from topological complement submodules p and q of E, ContinuousLinearMap.ofIsCompl is the induced continuous linear map E →L[R] F over the entire module. This is the continuous version of LinearMap.ofIsCompl.

Defined in

Mathlib.Topology.Algebra.Module.Complement

Cited by

10 results in Mathlib

Foundations

Depth 81 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

RingTopologicalSpaceAddCommGroupModuleIsTopologicalAddGroupTopologicalSpaceAddCommGroupModuleContinuousAdd