Transformation of P-Norm

Theorem
Let $p, q \geq 1$ be real numbers.

Let $\ell^p$ denote the $p$-sequence space.

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

Let $\mathbf{x} = \langle{x_n}\rangle \in \ell^{pq}$.

Suppose further that $\mathbf{x}^p = \langle{x_n^p}\rangle \in \ell^q$.

Then:


 * $\left\Vert \mathbf{x}^p \right\Vert_q = \left\Vert \mathbf{x} \right\Vert_{pq}^p$