How do you prove the Cauchy-Schwarz inequality?

How do you prove the Cauchy-Schwarz inequality?

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. Specifically, u · v = |u||v|cosθ, and cosθ ≤ 1.

What is Cauchy-Schwarz used for?

Taking square roots gives the triangle inequality: The Cauchy–Schwarz inequality is used to prove that the inner product is a continuous function with respect to the topology induced by the inner product itself.

Is Cauchy-Schwarz inequality a theorem?

Theorem 6.5 (Cauchy-Schwarz inequality) For any n-dimensional vectors u and v, u v ≤ u 2 v 2 , and equality occurs if and only if v = cu.

Why we use Cauchy-Schwarz inequality in statistics?

The Cauchy-Schwarz Inequality (also called Cauchy’s Inequality, the Cauchy-Bunyakovsky-Schwarz Inequality and Schwarz’s Inequality) is useful for bounding expected values that are difficult to calculate.

Why do we need Cauchy-Schwarz inequality?

The Cauchy-Schwarz inequality also is important because it connects the notion of an inner product with the notion of length. Show activity on this post. The Cauchy-Schwarz inequality holds for much wider range of settings than just the two- or three-dimensional Euclidean space R2 or R3.

What does the Cauchy-Schwarz inequality state?

What this is basically saying is that for two random variables, X and Y, the expected value of the square of them multiplied together E(XY)2 will always be less than or equal to the expected value of the product of the squares of each. E(X2)E(Y2).

What is hinge Theorem?

The Hinge Theorem states that if two sides of two triangles are congruent and the included angle is different, then the angle that is larger is opposite the longer side.

What is another name for hinge Theorem?

The SAS Inequality Theorem (informally known as the Hinge Theorem) states that BC>EF. The intuition for this theorem lies fully in its informal name. If a hinge is opened with a greater angle, then naturally the distance between the two ends is greater, even though the other side lengths are the same.

Does Cauchy-Schwarz hold for any norm?

The more familiar triangle inequality, that the length of any side of a triangle is bounded by the sum of the lengths of the other two sides is, in fact, an immediate consequence of the Cauchy–Schwarz inequality, and hence also valid for any norm based on an inner product.