Sequence space in the context of Natural numbers

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 ⁠${\displaystyle \mathbb {K} }$⁠ 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 ⁠${\displaystyle \mathbb {K} }$⁠, 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 ⁠${\displaystyle \textstyle \ell ^{p}}$⁠ spaces, consisting of the ⁠${\displaystyle p}$⁠-power summable sequences, with the ⁠${\displaystyle p}$⁠-norm. These are special cases of ⁠${\displaystyle L^{p}}$⁠ 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 ⁠${\displaystyle c}$⁠ and ⁠${\displaystyle c_{0}}$⁠, 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.