Transformation of P-Norm

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

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

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

Let $\mathbf x = \sequence {x_n} \in \ell^{p q}$.

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

Then:


 * $\norm {\mathbf x^p}_q = \norm {\mathbf x}_{p q}^p$