233 P-Norm
A norm defined by a parameter on a set of sequences that is used to find the distance between elements of the set.
definition [d] (\(p\)-Norm) From Kreyszig 2.2-3: on the space \(\ell^{p}\) the norm is given by
- \(\displaystyle \lVert x \rVert = \left( \sum_{j=1}^{\infty} |\xi_{j}|^{p} \right)^{1/p}\)
for \(x = (\xi_{j}) \in \ell^{p}\). This norm induces the metric of 1.2-3.
where
- \(p \geq 1\) is a fixed real number.
- \(x = (\xi_{j})\) is a sequence in \(\ell^{p}\).
- \(\lVert x \rVert\) is the \(p\)-norm of \(x\).
233.1 Elementary Example
233.1.1 Simple
The \(p\)-norm of a finite sequence uses the power \(p\) and the outer root \(1/p\). Take \(p = 1\).
\[ x = (1,-2,3),\quad p = 1 \]
\[ \lVert x \rVert = |1| + |-2| + |3| = 6 \]
where
- \(\lVert x \rVert\) is the \(p\)-norm of \(x\).
- \(p\) is the fixed exponent.
233.1.2 General
For \(p = 2\) on a longer sequence, square each absolute value, sum, then take the square root.
\[ x = (1,0,-1,2),\quad p = 2 \]
\[ \lVert x \rVert = \bigl(1^{2} + 0^{2} + (-1)^{2} + 2^{2}\bigr)^{1/2} = \sqrt{6} \]
where
- on \(\ell^{p}\) the same formula uses an infinite sum \(\sum_{j=1}^{\infty} |\xi_{j}|^{p}\).