Theorems · Definition · functional analysis

ContinuousMultilinearMap.curryFinFinset

(𝕜 : Type u) →
  (G : Type wG) →
    (G' : Type wG') →
      [inst : NontriviallyNormedField 𝕜] →
        [inst_1 : NormedAddCommGroup G] →
          [inst_2 : NormedSpace 𝕜 G] →
            [inst_3 : NormedAddCommGroup G'] →
              [inst_4 : NormedSpace 𝕜 G'] →
                {k l n : ℕ} →
                  {s : Finset (Fin n)} → s.card = k → sᶜ.card = l → (G [×n]→L[𝕜] G') ≃ₗᵢ[𝕜] G [×k]→L[𝕜] G [×l]→L[𝕜] G'

If s : Finset (Fin n) is a finite set of cardinality k and its complement has cardinality l, then the space of continuous multilinear maps G [×n]→L[𝕜] G' of n variables is isomorphic to the space of continuous multilinear maps G [×k]→L[𝕜] G [×l]→L[𝕜] G' of k variables taking values in the space of continuous multilinear maps of l variables.

Defined in

Mathlib.Analysis.Normed.Module.Multilinear.Curry

Cited by

11 results in Mathlib

Foundations

Depth 177 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

NontriviallyNormedFieldNormedAddCommGroupNormedSpaceNormedAddCommGroupNormedSpace