Table of Contents
What does the Cauchy-Schwarz inequality say?
In other words, the length of the sum of two vectors is no more then the sum of the lengths of the vectors. As explained in class, if you believe that vectors in hundreds of dimensions act like the vectors you know and love in R2, then the Cauchy-Schwartz inequality is a consequence of the law of cosines.
Why is Cauchy-Schwarz inequality useful?
The Cauchy-Schwarz inequality also is important because it connects the notion of an inner product with the notion of length. The Cauchy-Schwarz inequality holds for much wider range of settings than just the two- or three-dimensional Euclidean space R2 or R3.
Who discovered Cauchy-Schwarz inequality?
Cauchy-Schwarz inequality, Any of several related inequalities developed by Augustin-Louis Cauchy and, later, Herman Schwarz (1843–1921).
When Cauchy-Schwarz is equal?
Thus the Cauchy-Schwarz inequality is an equality if and only if u is a scalar multiple of v or v is a scalar multiple of u (or both; the phrasing has been chosen to cover cases in which either u or v equals 0).
What is the Cauchy-Schwarz relation for two vectors?
The Cauchy-Schwarz inequality applies to any vector space that has an inner product; for instance, it applies to a vector space that uses the L2-norm. u + v 2 ≤ u 2 + v 2 . The triangle inequality holds for any number of dimensions, but is easily visualized in ℝ3.
What is the Schwarz inequality in R 2 or R 3?
6.6 The Cauchy-Schwarz Inequality The Cauchy-Schwarz inequality is one of the most widely used inequalities in mathematics, and will have occasion to use it in proofs. We can motivate the result by assuming that vectors u and v are in ℝ2 or ℝ3. In either case, 〈u, v〉 = ‖u‖2‖v‖2 cos θ.
Does Cauchy-Schwarz hold for complex numbers?
The Cauchy-Schwarz-Bunjakowsky inequality in line (3. 1) holds in all complex vector spaces X, provided with a norm · and the product < ·|· > from Definition 1.1. Remark 3.2. This theorem is the main contribution of the paper.
What is Cauchy inequality in complex analysis?
Cauchy’s inequality may refer to: the Cauchy–Schwarz inequality in a real or complex inner product space. Cauchy’s inequality for the Taylor series coefficients of a complex analytic function.