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,

a1b1+...+anbn <= (a1p+...+anp)1/p(b1q+...+bnq)1/q

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
for f∈Lp and φ∈Lq.

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

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