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Cauchy–Schwarz inequality (Modified Schwarz inequality for 2-positive maps [27]) — For a 2-positive map between C*-algebras, for all , in its domain, () ‖ ‖ (), ‖ ‖ ‖ ‖ ‖ ‖. Another generalization is a refinement obtained by interpolating between both sides of the Cauchy–Schwarz inequality:
In mathematics, specifically in complex analysis, Cauchy's estimate gives local bounds for the derivatives of a holomorphic function. These bounds are optimal. These bounds are optimal. Cauchy's estimate is also called Cauchy's inequality , but must not be confused with the Cauchy–Schwarz inequality .
In mathematics, Cauchy's integral formula, named after Augustin-Louis Cauchy, is a central statement in complex analysis.It expresses the fact that a holomorphic function defined on a disk is completely determined by its values on the boundary of the disk, and it provides integral formulas for all derivatives of a holomorphic function.
The special case p = q = 2 gives a form of the Cauchy–Schwarz inequality. [1] 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 L p (μ) and g is in L q (μ), then the pointwise product fg is in L 1 (μ).
The Moore–Aronszajn theorem goes in the other direction; it states that every symmetric, positive definite kernel defines a unique reproducing kernel Hilbert space. The theorem first appeared in Aronszajn's Theory of Reproducing Kernels, although he attributes it to E. H. Moore. Theorem. Suppose K is a symmetric, positive definite kernel on a ...
The first pivotal theorem proved by Cauchy, now known as Cauchy's integral theorem, was the following: =, where f(z) is a complex-valued function holomorphic on and within the non-self-intersecting closed curve C (contour) lying in the complex plane. The contour integral is taken along the contour C. The rudiments of this theorem can already be ...
Many important inequalities can be proved by the rearrangement inequality, such as the arithmetic mean – geometric mean inequality, the Cauchy–Schwarz inequality, and Chebyshev's sum inequality.
Cauchy–Schwarz inequality – Mathematical inequality relating inner products and norms Hölder's inequality – Inequality between integrals in Lp spaces Mahler's inequality – inequality relating geometric mean of two finite sequences of positive numbers to the sum of each separate geometric mean Pages displaying wikidata descriptions as a ...