Theorems · Definition · functional analysis

lp.dualPairing

{ι : Type u_1} →
  {𝕜 : Type u_2} →
    {E : ι → Type u_3} →
      {F : ι → Type u_4} →
        [inst : RCLike 𝕜] →
          [inst_1 : (i : ι) → NormedAddCommGroup (E i)] →
            [inst_2 : (i : ι) → NormedSpace 𝕜 (E i)] →
              [inst_3 : (i : ι) → NormedAddCommGroup (F i)] →
                [inst_4 : (i : ι) → NormedSpace 𝕜 (F i)] →
                  (p q : ENNReal) →
                    {H : Type u_6} →
                      [inst_5 : NormedAddCommGroup H] →
                        [inst_6 : NormedSpace 𝕜 H] →
                          [CompleteSpace H] →
                            [inst_8 : Fact (1 ≤ p)] →
                              [inst_9 : Fact (1 ≤ q)] →
                                [p.HolderConjugate q] →
                                  (B : (i : ι) → E i →L[𝕜] F i →L[𝕜] H) →
                                    {K : NNReal} → (∀ (i : ι), ‖B i‖ ≤ ↑K) → ↥(lp E p) →L[𝕜] ↥(lp F q) →L[𝕜] H

The natural pairing between lp E p and lp F q (for Hölder conjugate p q : ℝ≥0∞) with values in a space H induced by a family of bilinear maps B : (i : ι) → E i →L[𝕜] F i →L[𝕜] H. This is given by ∑' i, B (e i) (f i). In the special case when B := (NormedSpace.inclusionInDoubleDual 𝕜 E).flip, which is definitionally the same as B := ContinuousLinearMap.id 𝕜 (E →L[𝕜] 𝕜), this is the natural map lp (fun _ ↦ StrongDual 𝕜 E) p →L[𝕜] StrongDual 𝕜 (lp E q).

Defined in

Mathlib.Analysis.Normed.Lp.lpHolder

Cited by

2 results in Mathlib

Foundations

Depth 231 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

RCLikeNormedAddCommGroupNormedSpaceNormedAddCommGroupNormedSpaceNormedAddCommGroupNormedSpaceCompleteSpaceFactFactENNReal.HolderConjugate