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Table of mathematical symbols

Lp space:

The space Lp is then defined as the set of all infinite sequences of real (or complex) numbers such that the p-norm is finite.

One can check that as p increases, the set Lp grows larger. For example, the sequence

$\left(1,\,\frac{1}{2},\,\cdots,\,\frac{1}{n},\,\frac{1}{n+1},\,\cdots\right)$

is not in L1, but it is in Lp for p>1, as the series

$1^p+\frac{1}{2^p}+\cdots+\frac{1}{n^p}+\frac{1}{(n+1)^p}+\cdots$

diverges for p=1 (the harmonic series), but is convergent for p>1.

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