Transformation of P-Norm

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

Let ${\ell^p}_\R$ denote the $p$-sequence space on $\R$.

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

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

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

Then:


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