P-Summable Sequence
A sequence of values whose elements determine a finite norm defined by a parameter that is used to identify members belonging to a matching set of sequences.
definition [d] (\(p\)-Summable Sequence) From Kreyszig 1.2-3: a sequence \(x = (\xi_{j}) = (\xi_{1}, \xi_{2}, \ldots)\) of numbers such that \(|\xi_{1}|^{p} + |\xi_{2}|^{p} + \cdots\) converges, that is
-
\(\displaystyle\sum_{j=1}^{\infty} |\xi_{j}|^{p} < \infty\) (\(p \geq 1\), fixed).
where
-
\(x = (\xi_{j})\) is the sequence.
-
\(p \geq 1\) is a fixed real number.
Elementary Example
Simple
A \(p\)-summable sequence is a sequence with \(\sum |\xi_{j}|^{p} < \infty\). For \(p = 1\) take three terms.
\[
x = (2,-1,4),\quad p = 1
\]
\[
\sum_{j=1}^{3} |\xi_{j}| = 7 < \infty
\]
where
-
\(x = (\xi_{j})\) is the \(p\)-summable sequence.
General
For \(p = 2\), zero-pad a four-term list to an infinite sequence in \(\ell^{2}\).
\[
x = (1,1,1,1,0,0,\ldots),\quad p = 2
\]
\[
\sum_{j=1}^{\infty} |\xi_{j}|^{2} = 4 < \infty
\]
where
- trailing zeros do not change the sum.
References
- Kreyszig, E. Introductory Functional Analysis with Applications. Wiley, 1989. — 1.2-3: sequences belonging to \(\ell^{p}\).