Any of several related inequalities developed by
The inequalities arise from assigning a {{link=real number">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, (∫fg) 2 /n≤/n(∫f 2 )(∫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, (Σ(a i , b i )) 2 ≤ Σ(a i ) 2 Σ(b i ) 2 . In addition to functional analysis , these inequalities have important applications in statistics and probability theory .