Cauchy-Schwarz Inequality

Tags: Augustin Cauchy, Bunyakovsky Inequality, Cauchy-Bunyakovsky-Schwarz Inequality, Cauchy-Schwarz Inequality, Hermann Amandus Schwarz, inequality, linear algebra, Math, probability theory, product space, Schwarz Inequality, square root, vectors, Viktor Yakovlevich Bunyakovsky

September 22, 2009

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Cauchy-Schwarz Inequality Explained

Description

A detailed tutorial on the solving of the Cauchy-Schwarz Inequality. Step by step tutorial including several examples of how to solve the Cauchy-Schwarz Inequality for reference.

Overview

The Cauchy-Schwarz Inequality is also known as the Schwarz Inequality, the Bunyakovsky Inequality, and the Cauchy-Bunyakovsky-Schwarz Inequality. It was published by Augustin Cauchy and was first stated by Viktor Yakovlevich Bunyakovsky. It was later rediscovered by Hermann Amandus Schwarz. This is used mostly in linear algebra, when solving vectors. It is also used in probability theory.

The Cauchy-Schwarz Inequality states that for all vectors x and y of an inner product space, $|\langle x,y\rangle|^2 \leq \langle x,x\rangle \cdot \langle y,y\rangle,$

By taking the square root of both sides, it can be written as $|\langle x,y\rangle| \leq \|x\| \cdot \|y\|.\,$

If the two sides are equal, and if x and y are both independent, then the formula may be restated as $\left|\sum_{i=1}^n x_i \overline{y_i}\right|^2 \leq \sum_{j=1}^n |x_j|^2 \sum_{k=1}^n |y_k|^2 .$

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Cauchy-Schwarz Inequality

Cauchy-Schwarz Inequality Explained

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