Hölder's Inequality for Sums/Formulation 2

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 $\sequence {x_n}_{n \mathop \in \N}$ and $\sequence {y_n}_{n \mathop \in \N}$ be (possibly infinite) convergent sequences in $\GF$.

Then:
 * $\ds \sum_{k \mathop \in \N} \size {x_k y_k} \le \paren {\sum_{k \mathop \in \N} \size {x_k}^p}^{1 / p} \paren {\sum_{k \mathop \in \N} \size {y_k}^q}^{1 / q}$

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:

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

Then:

By the Comparison Test, it follows that:

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