Cauchy-Schwarz inequality

Cauchy-Schwarz inequality

[kō·shē ′shwȯrts in·i′kwäl·əd·ē]
(mathematics)
The square of the inner product of two vectors does not exceed the product of the squares of their norms. Also known as Buniakowski's inequality; Schwarz' inequality.
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From a May 2016 conference in Chicago, 15 papers explore local and global methods in algebraic geometry from such perspectives as the canonical map of some surfaces isogenous to a product, the degeneration of differentials and moduli of nodal curves on K3 surfaces, algebraic fiber spaces over abelian varieties: around a recent theorem by Cao and Paun, full cones swept out by minimal rational curves on irreducible Hermitian symmetric spaces as examples of varieties underlying geometric substructures, and Skoda's ideal generation from vanishing theorem for semipositive Nakano curvature and Cauchy-Schwarz inequality for tensors.
From this equation, using the Cauchy-Schwarz inequality, we obtain that [parallel][[??].sup.n][parallel] [less than or equal to] [parallel][p.sup.n-1][parallel].
By the Hardy inequality [mathematical expression not reproducible] and by the Cauchy-Schwarz inequality, changing [xi] = [eta]y we find
Integrating (20) over (0, t), using Cauchy-Schwarz inequality and (18), we obtain for any [p.sub.1] > 0
Using [epsilon]-inequality and Cauchy-Schwarz inequality we obtain
Recalling the Cauchy-Schwarz inequality, we obtain the formula below:
Then, applying Cauchy-Schwarz inequality and (33), we obtain for any v [member of] W
By the Cauchy-Schwarz inequality and changing the order of integration, we have
From Cauchy-Schwarz inequality [absolute value of <[a.sub.i], [a.sub.j]>] [less than or equal to] [[parallel] [a.sub.i][parallel].sub.2] [[parallel][a.sub.J][parallel].sub.2], it is clear that the mutual coherence of any matrix A is bounded by 1.
In the 3D space [R.sup.(3)] for any couple vectors the Cauchy-Schwarz inequality is true that for the vectors (1) inequality gives an implication
Using the triangle inequality and Cauchy-Schwarz inequality, we have