C0 space | Wikipedia audio article
Channel: wikipedia tts
Date: 29/03/2019

This is an audio version of the Wikipedia Article:
https://en.wikipedia.org/wiki/Sequence_space


00:01:05 1 Definition
00:02:59 1.1 ℓsupip/i/sup spaces
00:03:54 1.2 ic/i, ic/isub0/sub and  ic/isub00/sub
00:04:49 1.3 Other sequence spaces
00:05:02 2 Properties of ℓsupip/i/sup spaces and the space ic/isub0/sub
00:06:43 2.1 ℓsupip/i/sup spaces are increasing in ip/i
00:10:15 3 See also
00:12:07 4 References
00:13:20 See also



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SUMMARY
=======
In functional analysis and related areas of mathematics, a sequence space is a vector space whose elements are infinite sequences of real or complex numbers. Equivalently, it is a function space whose elements are functions from the natural numbers to the field K of real or complex numbers.  The set of all such functions is naturally  identified with the set of all possible infinite sequences with elements in K, and can be turned into a vector space under the operations of pointwise addition of functions and pointwise scalar multiplication. All sequence spaces are linear subspaces of this space.  Sequence spaces are typically equipped with a norm, or at least the structure of a topological vector space.
The most important sequence spaces in analysis are the ℓp spaces, consisting of the p-power summable sequences, with the p-norm.  These are special cases of Lp spaces for the counting measure on the set of natural numbers.  Other important classes of sequences like convergent sequences or null sequences form sequence spaces, respectively denoted c and c0, with the sup norm. Any sequence space can also be equipped with the topology of pointwise convergence, under which it becomes a special kind of Fréchet space called FK-space.