Difference between revisions of "Cauchy-Schwarz Inequality"
(category) |
(→Elementary Form: parentheses) |
||
Line 6: | Line 6: | ||
<center> | <center> | ||
<math> | <math> | ||
− | \left( \sum_{i=1}^{n}a_ib_i \right)^2 \le \sum_{i=1}^{n}a_i^2 \sum_{i=1}^{n}b_i^2 | + | \left( \sum_{i=1}^{n}a_ib_i \right)^2 \le \left (\sum_{i=1}^{n}a_i^2 \right )\left (\sum_{i=1}^{n}b_i^2 \right ) |
</math>, | </math>, | ||
</center> | </center> |
Revision as of 21:25, 30 November 2007
The Cauchy-Schwarz Inequality (which is known by other names, including Cauchy's Inequality, Schwarz's Inequality, and the Cauchy-Bunyakovsky-Schwarz Inequality) is a well-known inequality with many elegant applications.
Contents
Elementary Form
For any real numbers and ,
,
with equality when there exist constants not both zero such that for all , .
Proof
There are several proofs; we will present an elegant one that does not generalize.
Consider the vectors and . If is the angle formed by and , then the left-hand side of the inequality is equal to the square of the dot product of and , or . The right hand side of the inequality is equal to . The inequality then follows from , with equality when one of is a multiple of the other, as desired.
Complex Form
The inequality sometimes appears in the following form.
Let and be complex numbers. Then
.
This appears to be more powerful, but it follows immediately from
.
General Form
Let be a vector space, and let be an inner product. Then for any ,
,
with equality if and only if there exist constants not both zero such that .
Proof 1
Consider the polynomial of
.
This must always be greater than or equal to zero, so it must have a non-positive discriminant, i.e., must be less than or equal to , with equality when or when there exists some scalar such that , as desired.
Proof 2
We consider
.
Since this is always greater than or equal to zero, we have
.
Now, if either or is equal to , then . Otherwise, we may normalize so that , and we have
,
with equality when and may be scaled to each other, as desired.
Examples
The elementary form of the Cauchy-Schwarz inequality is a special case of the general form, as is the Cauchy-Schwarz Inequality for Integrals: for integrable functions ,
,
with equality when there exist constants not both equal to zero such that for ,
.
Other Resources
Books
- The Cauchy-Schwarz Master Class: An Introduction to the Art of Mathematical Inequalities by J. Michael Steele.
- Problem Solving Strategies by Arthur Engel contains significant material on inequalities.