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Mathematics for Physics

359 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.

359.1 Examples

359.1.1 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.

359.1.2 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.