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What does the Cauchy-Schwarz inequality tell us?
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 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.
What is vector inequality?
Triangle Inequality in Vectors It is simply an expression of the fact that any side in a triangle is less than the sum of the other two sides, and greater than their difference.
Is there a way to prove the Cauchy-Schwarz inequality?
Proof of the Cauchy-Schwarz Inequality. There are various ways to prove this inequality. A short proof is given below. Consider the function. f ( x) = ( a 1 x − b 1) 2 + ( a 2 x − b 2) 2 + ⋯ + ( a n x − b n) 2. f (x)=\\left (a_1x-b_1\\right)^2+\\left (a_2 x-b_2\\right)^2+\\cdots +\\left (a_nx-b_n\\right)^2. f (x) = (a1. .
Which is an example of the Schwarz inequality?
The corresponding inequality for integrals was first proved by Hermann Schwarz ( 1888 ), he also gave the modern proof of the integral version. is the inner product. Examples of inner products include the real and complex dot product; see the examples in inner product.
When did Cauchy publish the inequality for sums?
The inequality for sums was published by Augustin-Louis Cauchy (1821). The corresponding inequality for integrals was first proved by Hermann Schwarz (1888), he also gave the modern proof of the integral version.
When does equality hold when two vectors are linearly dependent?
Equality hold if and only if the 2 vectors are linearly dependent. ∣ ∑ i = 1 n x i y ˉ i ∣ 2 ≤ ∑ j = 1 n ∣ x j ∣ 2 ∑ k = 1 n ∣ y k ∣ 2. ∣2. y y are linearly dependent, that is, one is a scalar multiple of the other (which includes the case when one or both are zero).