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- Hölder's_inequality abstract "In mathematical analysis Hölder's inequality, named after Otto Hölder, is a fundamental inequality between integrals and an indispensable tool for the study of Lp spaces.Theorem (Hölder's inequality). Let (S, Σ, μ) be a measure space and let p, q ∈ [1, ∞] with 1/p + 1/q = 1. Then, for all measurable real- or complex-valued functions f and g on S,If, in addition, p, q ∈ (1, ∞) and f ∈ Lp(μ) and g ∈ Lq(μ), then Hölder's inequality becomes an equality if and only if |f |p and |g |q are linearly dependent in L1(μ), meaning that there exist real numbers α, β ≥ 0, not both of them zero, such that α |f |p = β |g|q μ-almost everywhere.The numbers p and q above are said to be Hölder conjugates of each other. The special case p = q = 2 gives a form of the Cauchy–Schwarz inequality. Hölder's inequality holds even if ||fg||1 is infinite, the right-hand side also being infinite in that case. Conversely, if f is in Lp(μ) and g is in Lq(μ), then the pointwise product fg is in L1(μ).Hölder's inequality is used to prove the Minkowski inequality, which is the triangle inequality in the space Lp(μ), and also to establish that Lq(μ) is the dual space of Lp(μ) for p ∈ [1, ∞).Hölder's inequality was first found by Rogers (1888), and discovered independently by Hölder (1889).".
- Hölder's_inequality wikiPageID "191538".
- Hölder's_inequality wikiPageRevisionID "606314925".
- Hölder's_inequality first "L. P.".
- Hölder's_inequality id "H/h047514".
- Hölder's_inequality last "Kuptsov".
- Hölder's_inequality title "Hölder inequality".
- Hölder's_inequality subject Category:Articles_containing_proofs.
- Hölder's_inequality subject Category:Functional_analysis.
- Hölder's_inequality subject Category:Inequalities.
- Hölder's_inequality subject Category:Probabilistic_inequalities.
- Hölder's_inequality comment "In mathematical analysis Hölder's inequality, named after Otto Hölder, is a fundamental inequality between integrals and an indispensable tool for the study of Lp spaces.Theorem (Hölder's inequality). Let (S, Σ, μ) be a measure space and let p, q ∈ [1, ∞] with 1/p + 1/q = 1.".
- Hölder's_inequality label "Desigualdad de Hölder".
- Hölder's_inequality label "Desigualdade de Hölder".
- Hölder's_inequality label "Disuguaglianza di Hölder".
- Hölder's_inequality label "Hölder's inequality".
- Hölder's_inequality label "Hölder-Ungleichung".
- Hölder's_inequality label "Inégalité de Hölder".
- Hölder's_inequality label "Nierówność Höldera".
- Hölder's_inequality label "Ongelijkheid van Hölder".
- Hölder's_inequality label "Неравенство Гёльдера".
- Hölder's_inequality label "ヘルダーの不等式".
- Hölder's_inequality label "赫尔德不等式".
- Hölder's_inequality sameAs H%C3%B6lder's_inequality.
- Hölder's_inequality sameAs Hölderova_nerovnost.
- Hölder's_inequality sameAs Hölder-Ungleichung.
- Hölder's_inequality sameAs Desigualdad_de_Hölder.
- Hölder's_inequality sameAs Hölderren_desberdintza.
- Hölder's_inequality sameAs Inégalité_de_Hölder.
- Hölder's_inequality sameAs Disuguaglianza_di_Hölder.
- Hölder's_inequality sameAs ヘルダーの不等式.
- Hölder's_inequality sameAs 횔더_부등식.
- Hölder's_inequality sameAs Ongelijkheid_van_Hölder.
- Hölder's_inequality sameAs Nierówność_Höldera.
- Hölder's_inequality sameAs Desigualdade_de_Hölder.
- Hölder's_inequality sameAs Q731894.
- Hölder's_inequality sameAs Q731894.
- Hölder's_inequality wasDerivedFrom Hölder's_inequality?oldid=606314925.