What does the Cauchy Schwarz inequality say?

What does the Cauchy Schwarz inequality say?

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.

What condition is needed for equality to hold in the Cauchy Schwarz inequality?

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 sum of all 3 sides of a triangle?

When it has three line segments joined end to end. Thus, we can say that a triangle is a polygon, which has three sides, three angles, three vertices and the sum of all three angles of any triangle equals 180°.

Which of the following is the triangle inequality?

The triangle inequality states that the sum of the lengths of any two sides of a triangle is greater than the length of the remaining side. It follows from the fact that a straight line is the shortest path between two points. The inequality is strict if the triangle is non-degenerate (meaning it has a non-zero area).

Does Cauchy-Schwarz hold for complex numbers?

Theorem 3.1. 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. This new proof of the Cauchy-Schwarz inequality depends only on the norm in the vector space.

Why is Cauchy-Schwarz important?

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.

What is the rule for side lengths of a triangle?

According to the first triangle inequality theorem, the lengths of any two sides of a triangle must add up to more than the length of the third side. This means that you cannot draw a triangle that has side lengths 2, 7 and 12, for instance, since 2 + 7 is less than 12.

What is the rule for the third side of a triangle?

The Formula The Triangle Inequality Theorem states that the sum of any 2 sides of a triangle must be greater than the measure of the third side. Note: This rule must be satisfied for all 3 conditions of the sides.

What are the inequalities in one triangle?

The greater angle and greater side theorem states that within a triangle, longer sides lie opposite larger angles, and the triangle inequality theorem states that the sum of any two sides of a triangle must be greater than the third side.

Which is the proof of the Cauchy-Schwarz inequality?

Theorem 1 (The Cauchy-Schwarz Inequality): If then . Proof: Let . Then we want to prove that: Notice that the sum of squares is always nonnegative, and so for all we have that: Let , , and . Then: Suppose that . Then reduces to which is true.

How to prove Theorem 2 of triangle inequality?

Theorem 2 (The Triangle Inequality): If then . Proof: Let . Then: Square both sides of the equation and apply the Cauchy-Schwarz inequality at to get: as desired.

Which is the most important inequality in mathematics?

One of the most important inequalities in mathematics is inarguably the famous Cauchy-Schwarz inequality whose use appears in many important proofs. We will prove this important inequality and prove an analogue of the triangle inequality in higher dimension Euclidean $n$ -space.