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January 30, 2011

Two interesting proofs of the Cauchy-Schwarz inequality (real case)

Filed under: mathematics — ckrao @ 5:09 am

The Cauchy-Schwarz inequality is one of the most celebrated mathematical results known for its wide range of applications, as it holds in general inner product spaces.

\displaystyle \langle{x,y\rangle} \leq \left\|x\right\| \left\|y\right\|

or

\displaystyle \left(\sum_{i=1}^n x_i y_i\right)^2 \leq \left(\sum_{i=1}^n x_i^2\right)\left(\sum_{i=1}^n y_i^2\right)

Most notably, it is used in proving the triangle inequality for an inner product space (e.g. Euclidean space with the dot product), where the norm (e.g. length) is derived from the inner product.

In the real case, the most common proofs I have seen are either by looking at the discriminant of a non-negative quadratic form, or from Lagrange’s identity. Here are two more interesting proofs from [1] and [2] respectively.

1. (finite discrete case) We start with the following elementary inequality (assuming that x, y and x+y are non-zero):

\displaystyle \frac{(a+b)^2}{x+y} \leq \frac{a^2}{x} + \frac{b^2}{y}.

This can be shown to be equivalent to (ay-bx)^2 \geq 0.

By an easy proof by induction this can be extended to

\displaystyle \frac{(a_1 +a_2 + \ldots + a_n)^2}{b_1+b_2 + \ldots + b_n} \leq \frac{a_1^2}{b_1} + \frac{a_2^2}{b_2} + \ldots + \frac{a_n^2}{b_n}.

Now let a_i = x_i y_i and b_i = y_i^2 for i= 1, 2, \ldots, n. This magically leads to

\displaystyle \left(\sum_{i=1}^n x_i y_i\right)^2 \leq \left(\sum_{i=1}^n x_i^2\right)\left(\sum_{i=1}^n y_i^2\right),

as required.

2. (for any real inner product space) We simply use the non-negativity of the squared length of the difference of two normalised vectors x and y:

\displaystyle 0 \leq \left\|\frac{x}{||x||}- \frac{y}{||y||}\right\|^2 = 2 - 2\frac{\langle x, y \rangle}{\left\|x\right\| \left\|y\right\|}.

From this we obtain \displaystyle \langle{x,y\rangle} \leq \left\|x\right\| \left\|y\right\| as desired.

References

[1] “Cauchy-Schwarz Inequality: Yet Another Proof”, available at http://www-stat.wharton.upenn.edu/~steele/Publications/Books/CSMC/New%20Problems/CSNewProof/CauchySchwarzInequalityProof.pdf

[2] J. M. Steele, The Cauchy-Schwarz Master Class: An Introduction to the Art of Mathematical Inequalities, MAA Problem Books Series, 2004.

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