224 Ell P
A set of sequences of values where each sequence has a finite norm defined by a parameter that is used to find the distance between sequences.
definition [d] (\(\ell^{p}\)) From Kreyszig 1.2-3: let \(p \geq 1\) be a fixed real number. By definition, each element in the space \(\ell^{p}\) is a sequence \(x = (\xi_{j}) = (\xi_{1}, \xi_{2}, \ldots)\) of numbers such that \(|\xi_{1}|^{p} + |\xi_{2}|^{p} + \cdots\) converges; thus
- \(\displaystyle\sum_{j=1}^{\infty} |\xi_{j}|^{p} < \infty\) (\(p \geq 1\), fixed)
and the metric is defined by
- \(\displaystyle d(x,y) = \left( \sum_{j=1}^{\infty} |\xi_{j} - \eta_{j}|^{p} \right)^{1/p}\)
where \(y = (\eta_{j})\) and \(\sum |\eta_{j}|^{p} < \infty\).
From Kreyszig 2.2-3: the space \(\ell^{p}\) is a Banach space with norm
- \(\displaystyle \lVert x \rVert = \left( \sum_{j=1}^{\infty} |\xi_{j}|^{p} \right)^{1/p}\) .
where
- \(p \geq 1\) is a fixed real number.
- \(x = (\xi_{j})\) is a sequence of scalars.
- \(y = (\eta_{j})\) is another such sequence.
224.1 Elementary Example
224.1.1 Simple
The space \(\ell^{p}\) consists of \(p\)-summable sequences with the \(p\)-norm. For \(p = 1\), a three-term sequence is an element after zero-padding.
\[ p = 1,\quad x = (1,-2,3,0,0,\ldots) \]
\[ \lVert x \rVert = |1| + |-2| + |3| = 6 \]
where
- \(\ell^{p}\) is the set of all such sequences.
- \(\lVert x \rVert\) is the \(\ell^{p}\) norm.