Transformation of P-Norm

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


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


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