An extension of the famous Cauchy-Schwarz inequality, this inequality forms a cornerstone of analysis in p-normed Banach spaces.
Let 1/p+1/q=1 be conjugate exponents. Then for all nonnegative a1, ..., an and b1, ..., bn,
When p=q=2, we get the Cauchy-Schwarz inequality.
For 1<p<∞, the conjugate exponent specifies the conjugate space Lq=Lp* of continuous linear functionals. Thus Hölder's inequality gives the relationship
|φ(f)| <= ||φ||q||f||p
Use Hölder's inequality (among many other things) to prove Minkowski's inequality.