Cauchy's Inequality

Theorem

$\ds \sum {r_i}^2 \sum {s_i}^2 \ge \paren {\sum {r_i s_i} }^2$

where all of $r_i, s_i \in \R$.

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Proof 1

For any $\lambda \in \R$, we define $f: \R \to \R$ as the function:

$\ds \map f \lambda = \sum {\paren {r_i + \lambda s_i}^2}$

Now:

$\map f \lambda \ge 0$

because it is the sum of squares of real numbers.

Hence:

This is a quadratic equation in $\lambda$.

From Solution to Quadratic Equation:

$\ds a \lambda^2 + b \lambda + c = 0: a = \sum { {s_i}^2}, b = 2 \sum {r_i s_i}, c = \sum { {r_i}^2}$

The discriminant of this equation (that is $b^2 - 4 a c$) is:

$\ds D := 4 \paren {\sum {r_i s_i} }^2 - 4 \sum { {r_i}^2} \sum { {s_i}^2}$

Aiming for a contradiction, suppose $D$ is (strictly) positive.

Then $\map f \lambda = 0$ has two distinct real roots, $\lambda_1 < \lambda_2$, say.

From Sign of Quadratic Function Between Roots, it follows that $f$ is (strictly) negative somewhere between $\lambda_1$ and $\lambda_2$.

But we have:

$\forall \lambda \in \R: \map f \lambda \ge 0$

From this contradiction it follows that:

$D \le 0$

which is the same thing as saying:

$\ds \sum { {r_i}^2} \sum { {s_i}^2} \ge \paren {\sum {r_i s_i} }^2$

$\blacksquare$

Proof 2

From the Complex Number form of the Cauchy-Schwarz Inequality, we have:

$\ds \paren {\sum \cmod {w_i}^2} \paren {\sum \cmod {z_i}^2} \ge \cmod {\sum w_i z_i}^2$

where all of $w_i, z_i \in \C$.

As elements of $\R$ are also elements of $\C$, it follows that:

$\ds \sum \size {r_i}^2 \sum \size {s_i}^2 \ge \size {\sum r_i s_i}^2$

where all of $r_i, s_i \in \R$.

But from the definition of modulus, it follows that:

$\ds \forall r_i \in \R: \size {r_i}^2 = {r_i}^2$

Thus:

$\ds \sum {r_i}^2 \sum {s_i}^2 \ge \paren {\sum r_i s_i}^2$

where all of $r_i, s_i \in \R$.

$\blacksquare$

Also presented as

can also be expressed in the form:

$\ds \sum_{i \mathop = 1}^n r_i s_i \le \sqrt {\paren {\sum_{i \mathop = 1}^n {r_i}^2} \paren {\sum_{i \mathop = 1}^n {s_i}^2} }$

where all of $r_i, s_i \in \R$.

Vector Form

Let $\mathbf a$ and $\mathbf b$ be vectors in a vector space $V$.

Then:

$\paren {\mathbf a \cdot \mathbf b}^2 \le \paren {\mathbf a \cdot \mathbf a} \paren {\mathbf b \cdot \mathbf b}$

where $\cdot$ denotes dot product.

Also known as

is also known as the Cauchy-Schwarz Inequality.

However, that name is also given to the more general Cauchy-Bunyakovsky-Schwarz Inequality, and so will not be used in this context on $\mathsf{Pr} \infty \mathsf{fWiki}$.

Some sources refer to this as the Cauchy-Schwarz inequality for sums.

Source of Name

This entry was named for Augustin Louis Cauchy.

Historical Note

The result known as was first published by Augustin Louis Cauchy in $1821$.

It is a special case of the Cauchy-Bunyakovsky-Schwarz Inequality.

Sources

• 1964: Milton Abramowitz and Irene A. Stegun: Handbook of Mathematical Functions ... (previous) ... (next): $3$: Elementary Analytic Methods: $3.2$ Inequalities: Cauchy's Inequality: $3.2.9$
• 1965: Michael Spivak: Calculus on Manifolds ... (previous) ... (next): 1. Functions on Euclidean Space: Norm and Inner Product
• 1968: Murray R. Spiegel: Mathematical Handbook of Formulas and Tables ... (previous) ... (next): $\S 36$: Inequalities: Cauchy-Schwartz Inequality: $36.3$
• 1989: Ephraim J. Borowski and Jonathan M. Borwein: Dictionary of Mathematics ... (previous) ... (next): Cauchy's inequality
• 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): Cauchy-Schwarz inequality: $(2)$
• 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): Cauchy-Schwarz inequality: $(2)$
• 2009: Murray R. Spiegel, Seymour Lipschutz and John Liu: Mathematical Handbook of Formulas and Tables (3rd ed.) ... (previous) ... (next): $\S 37$: Inequalities: Cauchy-Schwartz Inequality: $37.3.$
• 2014: Christopher Clapham and James Nicholson: The Concise Oxford Dictionary of Mathematics (5th ed.) ... (previous) ... (next): Cauchy-Schwarz inequality for sums
• 2021: Richard Earl and James Nicholson: The Concise Oxford Dictionary of Mathematics (6th ed.) ... (previous) ... (next): Cauchy-Schwarz inequality for sums

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