Theorems · Definition · functional analysis

lp.holderL

{ι : Type u_1} →
  {𝕜 : Type u_2} →
    {E : ι → Type u_3} →
      {F : ι → Type u_4} →
        {G : ι → Type u_5} →
          [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)] →
                    [inst_5 : (i : ι) → NormedAddCommGroup (G i)] →
                      [inst_6 : (i : ι) → NormedSpace 𝕜 (G i)] →
                        {p q : ENNReal} →
                          (r : ENNReal) →
                            [hpqr : p.HolderTriple q r] →
                              [inst_7 : Fact (1 ≤ p)] →
                                [inst_8 : Fact (1 ≤ q)] →
                                  [inst_9 : Fact (1 ≤ r)] →
                                    (B : (i : ι) → E i →L[𝕜] F i →L[𝕜] G i) →
                                      {K : NNReal} → (∀ (i : ι), ‖B i‖ ≤ ↑K) → ↥(lp E p) →L[𝕜] ↥(lp F q) →L[𝕜] ↥(lp G r)

lp.holder as a continuous bilinear map.

Defined in

Mathlib.Analysis.Normed.Lp.lpHolder

Cited by

2 results in Mathlib

Foundations

Depth 230 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

RCLikeNormedAddCommGroupNormedSpaceNormedAddCommGroupNormedSpaceNormedAddCommGroupNormedSpaceENNReal.HolderTripleFactFactFact