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:

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

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

\(\ds \)

\(=\)

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

Definition of $p$-Norm

$\blacksquare$