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 $\GF \in \set {\R, \C}$, that is, $\GF$ represents the set of either the real numbers or the complex numbers.

Let $\mathbf x$ and $\mathbf y$ denote the sequences:
 * $\mathbf x = \sequence {x_n} \in {\ell^p}_\GF$
 * $\mathbf y = \sequence {y_n} \in {\ell^q}_\GF$

where:
 * ${\ell^p}_\GF$ denotes the complex $p$-sequence space in $\GF$:
 * $\ds {\ell^p}_\GF := \set {\sequence {z_n}_{n \mathop \in \N} \in \GF^\N: \sum_{n \mathop = 0}^\infty \size {z_n}^p < \infty}$
 * $\size {z_n}$ denotes either the modulus of $z_n \in \C$ or the absolute value of $z_n \in \R$ (the latter the restriction of the former to $\R$)
 * $\mathbf x$ and $\mathbf y$ are considered as vectors in the vector space $\F^\N$.

Let $\norm {\mathbf x}_p$ denote the $p$-norm of $\mathbf x$:
 * $\ds \norm {\mathbf x}_p = \paren {\sum_{n \mathop = 0}^\infty \size {x_n}^p}^{1 / p}$

Then $\mathbf x \mathbf y = \sequence {x_n y_n} \in \ell^1$, and:
 * $\norm {\mathbf x \mathbf y}_1 \le \norm {\mathbf x}_p \norm {\mathbf y}_q$

Proof
, assume that $\mathbf x$ and $\mathbf y$ are non-zero.

Define:
 * $\mathbf u = \sequence {u_n} = \dfrac {\mathbf x} {\norm {\mathbf x}_p}$
 * $\mathbf v = \sequence {v_n} = \dfrac {\mathbf y} {\norm {\mathbf y}_q}$

Then:
 * $\ds \norm {\mathbf u}_p = \dfrac 1 {\norm {\mathbf x}_p} \paren {\sum_{n \mathop = 0}^\infty \size {x_n}^p}^{1/p} = 1$

Similarly:
 * $\norm {\mathbf v}_q = 1$

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

By the comparison test, it follows that:
 * $\mathbf u \mathbf v = \sequence {u_n v_n} \in \ell^1$
 * $\mathbf x \mathbf y = \norm {\mathbf x}_p \norm {\mathbf y}_q \mathbf u \mathbf v \in \ell^1$

From $(1)$, it follows that:
 * $\norm {\mathbf u \mathbf v}_1 \le \dfrac 1 p \norm {\mathbf u}_p + \dfrac 1 q \norm {\mathbf v}_q = 1$

Therefore:
 * $\norm {\mathbf x \mathbf y}_1 = \norm {\mathbf x}_p \norm {\mathbf y}_q \norm {\mathbf u \mathbf v}_1 \le \norm {\mathbf x}_p \norm {\mathbf y}_q$

as desired.

Also see

 * Minkowski's Inequality for Sums