What is the point of the Cauchy-Schwarz 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.

Why is the triangle inequality called the triangle inequality?

These curves have different lengths, and only one—the one we call straight—has the shortest length. At the heart of this property lies the triangle inequality, which states that the sum of the lengths of any two sides of a triangle is greater than the length of the third side.

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 meant by triangle inequality?

triangle inequality, in Euclidean geometry, theorem that the sum of any two sides of a triangle is greater than or equal to the third side; in symbols, a + b ≥ c. In essence, the theorem states that the shortest distance between two points is a straight line.

What is Cauchy-Schwarz inequality example?

Example question: use the Cauchy-Schwarz inequality to find the maximum of x + 2y + 3z, given that x2 + y2 + z2 = 1. We know that: (x + 2y + 3x)2 ≤ (12 + 22 32)(x2 + y2 + z2) = 14. Therefore: x + 2y + 3z ≤ √14.

When triangle inequality is an equality?

In the Euclidean case, equality occurs only if the triangle has a 180° angle and two 0° angles, making the three vertices collinear, as shown in the bottom example. Thus, in Euclidean geometry, the shortest distance between two points is a straight line.

What is exterior angle inequality theorem?

The exterior angle inequality theorem states that the measure of any exterior angle of a triangle is greater than both of the non-adjacent interior angles.

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 relationship between the largest angle of a triangle and the side opposite it?

The longest side in a triangle is opposite the largest angle, and the shortest side is opposite the smallest angle. Triangle Inequality: In any triangle, the sum of the lengths of any two sides is greater than the length of the third side. Pythagorean Theorem: In a right triangle with hypotenuse .

Does Cauchy-Schwarz work for all norms?

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.

What is reverse triangle inequality?

Reverse triangle inequality states that the length of any side of the triangle is greater than the difference between the remaining two sides. Triangle inequality states that the sum of the lengths of two sides of the triangle is greater than, or equal to, the length of the remaining side.

What is a triangular 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.

What is triangle equality?

Triangle inequality, in Euclidean geometry , theorem that the sum of any two sides of a triangle is greater than or equal to the third side; in symbols, a + b ≥ c. In essence, the theorem states that the shortest distance between two points is a straight line.