NettetThe map defines a norm on (See Theorems 1 and 2 below.) The dual norm is a special case of the operator norm defined for each (bounded) linear map between normed vector spaces. Since the ground field of ( or ) is complete, is a Banach space. The topology on induced by turns out to be stronger than the weak-* topology on. Nettet14. jun. 2013 · Hint: Consider , , , . It is a fairly common inequality. Suppose that , then and and , so we can use the standard Hölder inequality to get Raising to the power yields. For I don't believe that it is correct, since the conjugate of would not fit. However, for I made a proof, I hope it is helpful to you.

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NettetThe reverse inequality follows from the same argument as the standard Minkowski, but uses that Holder's inequality is also reversed in this range. Using the Reverse … NettetHölder's inequality is a statement about sequences that generalizes the Cauchy-Schwarz inequality to multiple sequences and different exponents. Contents Proof Minkowski's … money characteristics

When does the equality hold in the Holder inequality?

NettetEquality holds when for all integers , i.e., when all the sequences are proportional. Statement If , , then and . Proof If then a.e. and there is nothing to prove. Case is … Hölder's inequality is used to prove the Minkowski inequality, which is the triangle inequality in the space L p (μ), and also to establish that L q (μ) is the dual space of L p (μ) for p ∈ [1, ∞). Hölder's inequality (in a slightly different form) was first found by Leonard James Rogers . Se mer In mathematical analysis, Hölder's inequality, named after Otto Hölder, is a fundamental inequality between integrals and an indispensable tool for the study of L spaces. The numbers p and q … Se mer Statement Assume that 1 ≤ p < ∞ and let q denote the Hölder conjugate. Then for every f ∈ L (μ), where max indicates that there actually is a g maximizing the … Se mer Statement Assume that r ∈ (0, ∞] and p1, ..., pn ∈ (0, ∞] such that $${\displaystyle \sum _{k=1}^{n}{\frac {1}{p_{k}}}={\frac {1}{r}}}$$ where 1/∞ is interpreted as 0 in this equation. Then for all … Se mer It was observed by Aczél and Beckenbach that Hölder's inequality can be put in a more symmetric form, at the price of introducing an extra vector (or function): Let Se mer Conventions The brief statement of Hölder's inequality uses some conventions. • In the definition of Hölder conjugates, 1/∞ means zero. Se mer For the following cases assume that p and q are in the open interval (1,∞) with 1/p + 1/q = 1. Counting measure For the n-dimensional Euclidean space, when the set S is {1, ..., n} with the counting measure, … Se mer Two functions Assume that p ∈ (1, ∞) and that the measure space (S, Σ, μ) satisfies μ(S) > 0. Then for all measurable real- or complex-valued functions f and g on S such that g(s) ≠ 0 for μ-almost all s ∈ S, Se mer NettetAbstract. In this paper, we shall prove that for n>1, the n-dimensional Jensen inequality holds for the g-expectation if and only if g is independent of y and linear with respect to z, in other ... money characteristics and functions