Sequence space

space

For , the space is a Banach space defined as the set of all sequences in with finite -norm given by

where in the case of we get the supremum

Thus is equivalent to the Lebesgue space ¹ where is the counting measure. More generally one defines with the counting measure for any set .

Proof of Banach space

This all follows from the general case of a Lebesgue space, however we will explicitly show completeness for .

Let be a Cauchy sequence in , i.e. for every there exists some such that for all we have

Now for any fixed

so is a Cauchy sequence in convergent to some . It remains to show that and . For any fixed

whence

and by the triangle inequality

hence

so . Furthermore

so indeed .

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Footnotes

1. There is no need to take a normed quotient here, is already a full norm due to properties of the counting measure. ↩