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:

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

Proof

\(\ds \norm {\mathbf x^p}_q\)

\(=\)

\(\ds \paren {\sum_{n \mathop = 0}^\infty \size { {x_n}^p}^q}^{1 / q}\)

Definition of $p$-Norm

\(\ds \)

\(=\)

\(\ds \paren {\sum_{n \mathop = 0}^\infty \size {x_n}^{p q} }^{1 / q}\)

Power of Power

\(\ds \)

\(=\)

\(\ds \paren {\paren {\sum_{n \mathop = 0}^\infty \size {x_n}^{p q} }^{1 / p q} }^p\)

Power of Power

\(\ds \)

\(=\)

\(\ds \norm {\mathbf x}_{p q}^p\)

Definition of $p$-Norm

$\blacksquare$