Table of Contents

1. Starting with Cauchy

Starting with Cauchy

Cauchy’s inequality for real numbers tells us that

$ \begin{equation} a_ {1} b_ {1}+a_ {2} b_ {2}+\cdots+a_ {n} b_ {n} \leq \sqrt{a_ {1}^{2}+a_ {2}^{2}+\cdots+a_ {n}^{2}} \sqrt{b_ {1}^{2}+b_ {2}^{2}+\cdots+b_ {n}^{2}} \end{equation} $

Thus, if neither of the sequences is made up of all zeros, we can introcduce new variables

$ \begin{equation} \hat{a}_ {k}=a_ {k} /\left(\sum_ {j} a_ {j}^{2}\right)^{\frac{1}{2}} \quad \text { and } \quad \hat{b}_ {k}=b_ {k} /\left(\sum_ {j} b_ {j}^{2}\right)^{\frac{1}{2}} \end{equation} $

which are normalized in the sense that

$ \begin{equation} \sum_ {k=1}^{\infty} \hat{a}_ {k}^{2}=\sum_ {k=1}^{\infty}\left\{a_ {k}^{2} /\left(\sum_ {j} a_ {j}^{2}\right)\right\}=1 \end{equation} $

and

$ \begin{equation} \sum_ {k=1}^{\infty} \hat{b}_ {k}^{2}=\sum_ {k=1}^{\infty}\left\{b_ {k}^{2} /\left(\sum_ {j} b_ {j}^{2}\right)\right\}=1 \end{equation} $

we obtain the simple-looking bound

$ \begin{equation} \sum_ {k=1}^{\infty} \hat{a}_ {k} \hat{b}_ {k} \leq \frac{1}{2} \sum_ {k=1}^{\infty} \hat{a}_ {k}^{2}+\frac{1}{2} \sum_ {k=1}^{\infty} \hat{b}_ {k}^{2}=1 \end{equation} $

and, in terms of the original sequence {a_k } and {b_k }, we have

$ \begin{equation} \sum_ {k=1}^{\infty}\left\{a_ {k} /\left(\sum_ {j} a_ {j}^{2}\right)^{\frac{1}{2}}\right\}\left\{b_ {k} /\left(\sum_ {j} b_ {j}^{2}\right)^{\frac{1}{2}}\right\} \leq 1 \end{equation} $

Finally, when we clear the denominators, we find out old friend Cauchy’s inequality:

$ \begin{equation} \sum_ {k=1}^{\infty} a_ {k} b_ {k} \leq\left(\sum_ {j=1}^{\infty} a_ {j}^{2}\right)^{\frac{1}{2}}\left(\sum_ {j=1}^{\infty} b_ {j}^{2}\right)^{\frac{1}{2}} \end{equation} $

We often benefit from the introduction of shorthand notation such as

$ \begin{equation} \langle\mathbf{a}, \mathbf{b}\rangle=\sum_ {j=1}^{n} a_ {j} b_ {j} \end{equation} $

This shorthand now permits us to write Cauchy’s inequality quite succinctly as

$ \begin{equation} \langle\mathbf{a}, \mathbf{b}\rangle \leq\langle\mathbf{a}, \mathbf{a}\rangle^{\frac{1}{2}}\langle\mathbf{b}, \mathbf{b}\rangle^{\frac{1}{2}} \end{equation} $

Specifically, if V is a real vector space (such as $ \mathbb{R}^{d} $ ), then we say that a function on V x V defined by the mapping (a, b) -> <a, b> is an inner product and we say that (V, <.,.>) is a real inner product space provided that the pair (V, <.,.>) has the following five properties:

(i) <v, v> >= 0 for all $ v \in V $

(ii) <v, v> = 0 if and only if v = 0

(iii) $ \langle\alpha \mathbf{v}, \mathbf{w}\rangle=\alpha\langle\mathbf{v}, \mathbf{w}\rangle $ for all $ \alpha \in R $ and all $ v, w \in V $

(iv) $ \langle\mathbf{u}, \mathbf{v}+\mathbf{w}\rangle=\langle\mathbf{u}, \mathbf{v}\rangle+\langle\mathbf{u}, \mathbf{w}\rangle $ for all $ u, v, w \in V $

(v) $ \langle\mathbf{v}, \mathbf{w}\rangle=\langle\mathbf{w}, \mathbf{v}\rangle $ for all $ v, w \in V $

For any real inner product space (V, <.,.>), one has for all v and w in V that

$ \begin{equation} \langle\mathbf{v}, \mathbf{w}\rangle \leq\langle\mathbf{v}, \mathbf{v}\rangle^{\frac{1}{2}}\langle\mathbf{w}, \mathbf{w}\rangle^{\frac{1}{2}} \end{equation} $

moreover, for nonzero vectors v and w, one has

$ \begin{equation} \langle\mathbf{v}, \mathbf{w}\rangle=\langle\mathbf{v}, \mathbf{v}\rangle^{\frac{1}{2}}\langle\mathbf{w}, \mathbf{w}\rangle^{\frac{1}{2}} \quad \text { if and only } i f \mathbf{v}=\lambda \mathbf{w} \end{equation} $

for a nonzero constant $ \lambda $

A great many of those applications depend on a natural analog of Cauchy’s inqeuality where sums are replaced by integrals,

$ \begin{equation} \int_ {a}^{b} f(x) g(x) d x \leq\left(\int_ {a}^{b} f^{2}(x) d x\right)^{\frac{1}{2}}\left(\int_ {a}^{b} g^{2}(x) d x\right)^{\frac{1}{2}} \end{equation} $