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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:
Proof. First equation: ... be a random variable with probability density function (; ... The Cauchy–Schwarz inequality shows that ...
The method can also be used on distributional limits of random variables. Furthermore, the estimate of the previous theorem can be refined by means of the so-called Paley–Zygmund inequality. Suppose that X n is a sequence of non-negative real-valued random variables which converge in law to a random variable X.
Lagrange's identity for complex numbers has been obtained from a straightforward product identity. A derivation for the reals is obviously even more succinct. Since the Cauchy–Schwarz inequality is a particular case of Lagrange's identity, [4] this proof is yet another way to obtain the CS inequality. Higher order terms in the series produce ...
The proof involves taking a multivariate random variable with density function and adding a location parameter to form a family of densities {()}. Then, by analogy with the Minkowski–Steiner formula , the "surface area" of X {\displaystyle X} is defined to be
A random vector = (, …,) is said to have the multivariate Cauchy distribution if every linear combination of its components = + + has a Cauchy distribution. That is, for any constant vector a ∈ R k {\displaystyle a\in \mathbb {R} ^{k}} , the random variable Y = a T X {\displaystyle Y=a^{T}X} should have a univariate Cauchy distribution. [ 29 ]
In mathematics, the following inequality is known as Titu's lemma, Bergström's inequality, Engel's form or Sedrakyan's inequality, respectively, referring to the article About the applications of one useful inequality of Nairi Sedrakyan published in 1997, [1] to the book Problem-solving strategies of Arthur Engel published in 1998 and to the book Mathematical Olympiad Treasures of Titu ...
In mathematics, an inner product space (or, rarely, a Hausdorff pre-Hilbert space [1] [2]) is a real vector space or a complex vector space with an operation called an inner product. The inner product of two vectors in the space is a scalar , often denoted with angle brackets such as in a , b {\displaystyle \langle a,b\rangle } .