# What is Cauchy Schwarz equation?

## What is Cauchy Schwarz equation?

The Cauchy-Schwarz inequality, also known as the Cauchy–Bunyakovsky–Schwarz inequality, states that for all sequences of real numbers a i a_i ai and b i b_i bi, we have. ( ∑ i = 1 n a i 2 ) ( ∑ i = 1 n b i 2 ) ≥ ( ∑ i = 1 n a i b i ) 2 .

## Is Cauchy-Schwarz inequality 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.

Is Cauchy Schwarz inequality important for JEE?

There are many reformulations of this inequality. There is a vector form and a complex number version too. But we only need the elementary form to tackle the problems. So, Cauchy Schwarz Inequality is useful in solving problems at JEE Level.

What vector space does the Cauchy-Schwarz inequality apply to?

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. Recall in high school geometry you were told that the sum of the lengths of two sides of a triangle is greater than the third side.

### What are some real life examples of Cauchy-Schwarz?

The following is one of the most common examples of the use of Cauchy-Schwarz. We can easily generalize this approach to show that if x^2 + y^2 + z^2 = 1 x2 + y2 +z2 = 1, then the maximum value of ax + by + cz ax+by +cz is

### How do you apply Cauchy-Schwarz to the RHS?

At first glance, it is not clear how we can apply Cauchy-Schwarz, as there are no squares that we can use. Furthermore, the RHS is not a perfect square. The power of Cauchy-Schwarz is that it is extremely versatile, and the right choice of can simplify the problem. ( a c × c + b a × a + c b × b) 2 ≤ ( a 2 c + b 2 a + c 2 b) ( c + a + b).

Does holders inequality generalize Cauchy-Schwarz?

We can also derive the Cauchy-Schwarz inequality from the more general Hölder’s inequality. Simply put r = 2 r = 2, and we arrive at Cauchy Schwarz. As such, we say that Holders inequality generalizes Cauchy-Schwarz.

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