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 a_{1}, ..., a_{n} and b_{1}, ..., b_{n},

a_{1}b_{1}+...+a_{n}b_{n} <=
(a_{1}^{p}+...+a_{n}^{p})^{1/p}(b_{1}^{q}+...+b_{n}^{q})^{1/q}

When p=q=2, we get the Cauchy-Schwarz inequality.

For 1<p<∞, the conjugate exponent specifies the conjugate space L_{q}=L_{p}^{*} of continuous linear functionals. Thus Hölder's inequality gives the relationship

|φ(f)| <= ||φ||_{q}||f||_{p}

for f∈L

_{p} and φ∈L

_{q}.

Use Hölder's inequality (among many other things) to prove Minkowski's inequality.