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, ..., a_n and b₁, ..., b_n,

a₁b₁+...+a_nb_n <= (a₁^p+...+a_n^p)^1/p(b₁^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

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