Transformation of P-Norm
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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 |
$\blacksquare$