Cauchy-Schwarz and triangle inequalities

Cauchy-Schwarz and triangle inequalities

Arun Ram
Department of Mathematics and Statistics
University of Melbourne
Parkville, VIC 3010 Australia
aram@unimelb.edu.au

Last updates: 03 January 2012

The space n

Define n={ (x1,x2 ,,xn) | x1,x2 ,,xn } so that 1=, 2= {(x,y) | x,y} , 3= {(x,y,z) | x,y,z} .

Creator: FreeHEP Graphics2D Driver Producer: geogebra.d.b Revision: 1.10 Source: Date: Monday, 7 December 2009 4:26:11 PM EST
Fig. Examples of points in 3

The absolute value on n is the function |x| : n 0 x |x| given by |x| = x12 ++ xn2 for x= ( x1,,xn ) .

For example:

Lagrange's identity

(Lagrange's identity) ( i=1n xi2 ) ( i=1n yi2 ) - ( i=1n xiyi ) 2 = 12 i,j=1 n ( xiyj -xjyi )2 .

Proof.

For example, when n=2, OOPS THIS IS MESSED UP SOMEHOW 12 (( x1y1 -x1y1 )2 +( x1y2 -x2y1) 2 +( x2y1 -x1y2) 2 + (x2y2 - x2y2) 2) = (x1y2 -x2y1) 2 = x12 y22 - 2x1x2 y1y2 + x22 y12 = (x12 +x22) (y12 + y22) -( x1y1 +x2 y2)2 .

The inner product

The inner product on n is the function n×n (x,y) x,y given by x,y = (x1, xn) ( y1 yn ) = x1y1 ++ xnyn = i=1n xiyi .

Note: The length of xn is given by |x| = x12+ +xn2 = x,x .

(The Cauchy-Schwarz inequality) Let x,yn. Then x,y |x| |y| .

Proof.

(The triangle inequality) Let x,y n. Then |x+y| |x|+|y| .

Proof.