Theorems · Definition · functional analysis

lp.holder

{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] →
                              (B : (i : ι) → E i →L[𝕜] F i →L[𝕜] G i) →
                                {K : ℝ} → (∀ (i : ι), ‖B i‖ ≤ K) → ↥(lp E p) → ↥(lp F q) → ↥(lp G r)

The map between lp spaces satisfying ENNReal.HolderTriple induced by a uniformly bounded family of continuous bilinear maps on the underlying spaces.

Defined in

Mathlib.Analysis.Normed.Lp.lpHolder

Cited by

2 results in Mathlib

Foundations

Depth 215 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

RCLikeNormedAddCommGroupNormedSpaceNormedAddCommGroupNormedSpaceNormedAddCommGroupNormedSpaceENNReal.HolderTriple