Any of several related inequalities developed by Augustin-Louis Cauchy and, later, Herman Schwarz (18431921).
The inequalities arise from assigning a real number measurement, or norm, to the function s, vector s, or integral s within a particular space in order to analyze their relationship. For functions f and g , whose squares are integrable and thus usable as a norm, ( 222B; fg) 2 /n 2264; /n( 222B; f 2 )( 222B; g 2 ). For vectors a = (a 1 , a 2 , a 3 ,..., a n ) and b = (b 1 , b 2 , b 3 ,..., b n ), together with the inner product (see inner product space ) for a norm, ( 03A3; (a i , b i )) 2 2264; 03A3; (a i ) 2 03A3; (b i ) 2 . In addition to functional analysis , these inequalities have important applications in statistics and probability theory .