User:Addem/Holder

Hölder's Inequality
There are two theorems which go by the title of Hölder's Inequality, one dealing with summation and the other with integration with respect to a measure.

In both cases it is asserted that, when $1 \le p \le \infty$ and $q$ is the exponential conjugate of $p$, then


 * $\norm {a b}_1 \le \norm a_p \norm b_q$

In the summation case we interpret $\norm {\, \cdot \,}_p$ as the $p$-norm and in the integral case we interpret it as the $p$-seminorm.

The result for summation can be deduced from the result for integration, using the counting measure. It also has independent proofs.