Theorems · Definition · functional analysis

ContinuousMultilinearMap.toContinuousLinearMap

{R : Type u} →
  {ι : Type v} →
    {M₁ : ι → Type w₁} →
      {M₂ : Type w₂} →
        [inst : Semiring R] →
          [inst_1 : (i : ι) → AddCommMonoid (M₁ i)] →
            [inst_2 : AddCommMonoid M₂] →
              [inst_3 : (i : ι) → Module R (M₁ i)] →
                [inst_4 : Module R M₂] →
                  [inst_5 : (i : ι) → TopologicalSpace (M₁ i)] →
                    [inst_6 : TopologicalSpace M₂] →
                      ContinuousMultilinearMap R M₁ M₂ → [DecidableEq ι] → ((i : ι) → M₁ i) → (i : ι) → M₁ i →L[R] M₂

If f is a continuous multilinear map, then f.toContinuousLinearMap m i is the continuous linear map obtained by fixing all coordinates but i equal to those of m, and varying the i-th coordinate.

Defined in

Mathlib.Topology.Algebra.Module.Multilinear.Basic

Cited by

11 results in Mathlib

Foundations

Depth 79 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

SemiringAddCommMonoidAddCommMonoidModuleModuleTopologicalSpaceTopologicalSpaceDecidableEq