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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 the estimation lemma, also known as the ML inequality, gives an upper bound for a contour integral. If f is a complex -valued, continuous function on the contour Γ and if its absolute value | f ( z ) | is bounded by a constant M for all z on Γ , then
The real absolute value function is an example of a continuous function that achieves a global minimum where the derivative does not exist. The subdifferential of | x | at x = 0 is the interval [−1, 1]. [14] The complex absolute value function is continuous everywhere but complex differentiable nowhere because it violates the Cauchy–Riemann ...
Bennett's inequality, an upper bound on the probability that the sum of independent random variables deviates from its expected value by more than any specified amount Bhatia–Davis inequality , an upper bound on the variance of any bounded probability distribution
The absolute value | | is a norm on the vector space formed by the real or complex numbers. The complex numbers form a one-dimensional vector space over themselves and a two-dimensional vector space over the reals; the absolute value is a norm for these two structures.
The standard absolute value on the integers. The standard absolute value on the complex numbers.; The p-adic absolute value on the rational numbers.; If R is the field of rational functions over a field F and () is a fixed irreducible polynomial over F, then the following defines an absolute value on R: for () in R define | | to be , where () = () and ((), ()) = = ((), ()).
The first of these quadratic inequalities requires r to range in the region beyond the value of the positive root of the quadratic equation r 2 + r − 1 = 0, i.e. r > φ − 1 where φ is the golden ratio. The second quadratic inequality requires r to range between 0 and the positive root of the quadratic equation r 2 − r − 1 = 0, i.e. 0 ...
The second inequality is the elementary inequality between and . The last inequality follows by applying reverse Fatou lemma , i.e. applying the Fatou lemma to the non-negative functions g − f n {\displaystyle g-f_{n}} , and again (up to sign) cancelling the finite ∫ X g d μ {\displaystyle \int _{X}g\,d\mu } term.
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