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### Cauchy-Schwarz Inequality

```
Date: 11/13/1999 at 23:36:15
From: Nick Krupar
Subject: Cauchy-Schwarz Inequality

Prove:
n                    n              n
[sum(a_j)(b_j)]^2 <= (sum[j(a_j)^2])(sum[[(b_j)^2]/j])
j=1                  j=1            j=1

I've shown that, where <...> is the inner-product:

n
[sum(a_j)(b_j)]^2 = <(a_1, ..., a_n),(b_1, ..., b_n)>^2
j=1

n             n
<= (sum[(a_j)^2])(sum[(b_j)^2])
j=1           j=1

I know that:

n               n
sum[(a_j)^2] <= sum[j(a_j)^2]
j=1             j=1

n               n
sum[(b_j)^2] >= sum[[(b_j)^2]/j]
j=1             j=1
```

```
Date: 11/14/1999 at 08:17:31
From: Doctor Anthony
Subject: Re: Cauchy-Schwarz Inequality

The Cauchy-Schwarz inequality is that for real numbers ai, bi then

(a1.b1 + a2.b2 + .... + an.bn)^2 <= (a1^2 + a2^2 + ... + an^2)*
(b1^2 + b2^2 + ... + bn^2)

Proof:

For any x we have

0 <= (a1.x+b1)^2 + (a2.x+b2)^2 + ... + (an.x + bn)^2

= (a1^2 + ... + an^2)x^2 + 2(a1.b1 + ... + an.bn)x +
(b1^2 + ... + bn^2)
=  Ax^2 + 2Bx + C

since y = Ax^2 + 2Bx + C >= 0 for ALL x it follows that the equation
Ax^2 + 2Bx + C = 0 cannot have two distinct roots. And so

(2B)^2 <= 4AC    (discriminant of the quadratic equation <= 0)
so
B^2 <= AC
so
(a1.b1 + ... + an.bn)^2 <= (a1^2 + ... + an^2)(b1^2 + ... + bn^2)

which is the required inequality.

A further inequality that is often quoted is:

(x1 + x2 + x3 + ...+ xn)^2 <= n(x1^2 + x2^2 + ... + xn^2)

If we take just 3 values, we require to show

(x1 + x2 + x3)^2 <= 3(x1^2 + x2^2 + x3^2)

x1^2+x2^2+x3^2+2x1.x2+2x2.x3+2x3.x1 <= 3(x1^2 + x2^2 + x3^2)

and so

2x1.x2 + 2x2.x3 + 2x3.x1 <= 2x1^2 + 2x2^2 + 2x3^2

0 <= x1^2 - 2x1.x2 + x^2 + x1^2 - 2x1.x3 + x3^2 + x2^2 - 2x2.x3
+ x3^2
0 <= (x1-x2)^2 + (x1-x3)^2 + (x2-x3)^2

which is clearly true since each term on the right is >= 0

The same pattern is true if you take higher values of n. You will
always be able to put the inequality into the form shown above.

- Doctor Anthony, The Math Forum
http://mathforum.org/dr.math/
```
Associated Topics:
College Number Theory

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