Hölder's Inequality for Sums

Theorem
Let $p, q \in \R_{>0}$ be strictly positive real numbers such that:
 * $\dfrac 1 p + \dfrac 1 q = 1$

Let:
 * $\mathbf{x} = \left\langle {x_n} \right\rangle \in \ell^p$
 * $\mathbf{y} = \left\langle {y_n} \right\rangle \in \ell^q$

where $\ell^p$ denotes the $p$-sequence space.

Let $\left\Vert {\mathbf{x}} \right\Vert_p$ denote the $p$-norm of $\mathbf{x}$.

Then $\mathbf{x} \mathbf{y} = \left\langle {x_n y_n} \right\rangle \in \ell^1$, and:
 * $\left\Vert {\mathbf{x} \mathbf{y}} \right\Vert_1 \le \left\Vert {\mathbf{x}} \right\Vert_p \left\Vert {\mathbf{y}} \right\Vert_q$

Proof
Assume WLOG that $\mathbf x$ and $\mathbf y$ are non-zero.

Define:
 * $\mathbf u = \langle {u_n} \rangle = \dfrac {\mathbf x} {\left\Vert {\mathbf x} \right\Vert_p}$
 * $\mathbf v = \langle {v_n} \rangle = \dfrac {\mathbf y} {\left\Vert {\mathbf y} \right\Vert_q}$

Then:
 * $\displaystyle \left\Vert {\mathbf u} \right\Vert_p = \dfrac {1} {\left\Vert {\mathbf x} \right\Vert_p} \left({ \sum_{n \mathop = 0}^\infty \left\vert{x_n}\right\vert^p }\right)^{1/p} = 1$

Similarly:
 * $\left\Vert {\mathbf{v}} \right\Vert_q = 1$

By Young's Inequality for Products:
 * $(*): \quad \forall n \in \N: \left\vert u_n v_n \right\vert \le \dfrac 1 p \left\vert u_n \right\vert^p + \dfrac 1 q \left\vert v_n \right\vert^q$

By the comparison test, it follows that:
 * $\mathbf{u} \mathbf{v} = \left\langle{u_n v_n}\right\rangle \in \ell^1$
 * $\mathbf{x} \mathbf{y} = \left\Vert {\mathbf x} \right\Vert_p \left\Vert {\mathbf y} \right\Vert_q \mathbf{u} \mathbf{v} \in \ell^1$

From $(*)$, it follows that:
 * $\left\Vert {\mathbf{u} \mathbf{v}} \right\Vert_1 \le \dfrac 1 p \left\Vert {\mathbf{u}} \right\Vert_p + \dfrac 1 q \left\Vert {\mathbf{v}} \right\Vert_q = 1$

Therefore:
 * $\left\Vert {\mathbf{x} \mathbf{y}} \right\Vert_1 = \left\Vert {\mathbf x} \right\Vert_p \left\Vert {\mathbf y} \right\Vert_q \left\Vert {\mathbf{u} \mathbf{v}} \right\Vert_1 \le \left\Vert {\mathbf x} \right\Vert_p \left\Vert {\mathbf y} \right\Vert_q$

as desired.

Also see

 * Minkowski's Inequality for Sums

It was first found by L. J. Rogers in 1888, and discovered independently by Hölder in 1889.