Hölder's Inequality for Sums/Formulation 1/Equality

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

Let:
 * $\mathbf x = \sequence {x_n} \in \ell^p$
 * $\mathbf y = \sequence {y_n} \in \ell^q$

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

Let $\norm {\mathbf x}_p$ denote the $p$-norm of $\mathbf x$.

Hölder's Inequality for Sums states that:
 * $\mathbf x \mathbf y \in \ell^1$

and:
 * $\norm {\mathbf x \mathbf y}_1 \le \norm {\mathbf x}_p \norm {\mathbf y}_q$

We have that:
 * $\norm {\mathbf x \mathbf y}_1 = \norm {\mathbf x}_p \norm {\mathbf y}_q$


 * $\forall k \in \N: \size {y_k} = c \size {x_k}^{p - 1}$
 * $\forall k \in \N: \size {y_k} = c \size {x_k}^{p - 1}$

for some real constant $c$.