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In functional analysis and related areas of mathematics, a sequence space is an important class of function space. Functional analysis is that branch of mathematics and specifically of analysis which is concerned with the study of spaces of functions. ...
Euclid, detail from The School of Athens by Raphael. ...
In mathematics, a function space is a set of functions of a given kind from a set X to a set Y. It is called a space because in many applications, it is a topological space or a vector space or both. ...
The set of all functions from the natural numbers to complex numbers, which can naturally be identified with the set of all possible infinite sequences with elements in  , can be turned into a vector space. Any linear subspace of this space is then called sequence space. Natural number can mean either a positive integer (1, 2, 3, 4, ...) or a non-negative integer (0, 1, 2, 3, 4, ...). Natural numbers have two main purposes: they can be used for counting (there are 3 apples on the table), or they can be used for ordering (this is...
The complex numbers are an extension of the real numbers, in which all non-constant polynomials have roots. ...
This is a page about mathematics. ...
Vector spaces (or linear spaces) are spaces whose elements, known as vectors, can be scaled and added; all linear combinations can be formed. ...
The concept of a linear subspace (or vector subspace) is important in linear algebra and related fields of mathematics. ...
Many important classes of sequences like bounded sequences or null sequences form sequence spaces. A sequence space equipped with the topology of pointwise convergence becomes a special kind of Fréchet space called FK-space. In mathematics, a function f defined on some set X with real or complex values is called bounded, if the set of its values is bounded. ...
Topology (Greek topos, place and logos, study) is a branch of mathematics concerned with spatial properties preserved under bicontinuous deformation (stretching without tearing or gluing); these are the topological invariants. ...
Suppose { fn } is a sequence of functions sharing the same domain in common (for the moment, we defer making precise the nature of the values of these functions, but the reader may take them to be real numbers if that makes anyone feel good). ...
This article deals with Fréchet spaces in functional analysis. ...
In functional analysis and related areas of mathematics a FK-space or Fréchet coordinate space is a sequence space equipped with a topological structure such that it becomes a Fréchet space. ...
Definition
We identify the set of all functions  with the set of all sequences  with  This set can be turned into a vector space by defining vector addition as Vector spaces (or linear spaces) are spaces whose elements, known as vectors, can be scaled and added; all linear combinations can be formed. ...
A vector in physics and engineering typically refers to a quantity that has close relationship to the spatial coordinates, informally described as an object with a magnitude and a direction. The word vector is also now used for more general concepts (see also vector and generalizations below), but in this...
 and the scalar multiplication as In mathematics, scalar multiplication is one of the basic operations defining a vector space in linear algebra (or more generally, a module in abstract algebra). ...
 A sequence space X is a linear subspace of ω.
Examples The space of bounded sequence  (sometimes called m) consisting of all bounded sequences In mathematics, a function f defined on some set X with real or complex values is called bounded, if the set of its values is bounded. ...
 The space of convergent sequences c consisting of all convergent sequences Convergence means approaching a definite value, as time goes on; or approaching a definite point, or a common view or opinion, or a fixed state of affairs. ...
 The space of null sequences c0 consisting of all null sequences  The space of finite sequences Φ consisting of all sequences where only a finite number of terms are non-zero.
The space of bounded series bs
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