In the mathematical field of

functional analysis Functional analysis is a branch of mathematical analysis, the core of which is formed by the study of vector spaces endowed with some kind of limit-related structure (e.g. Inner product space#Definition, inner product, Norm (mathematics)#Defini ...

, the space denoted by ''c'' is the

vector space In mathematics and physics, a vector space (also called a linear space) is a set whose elements, often called ''vectors'', may be added together and multiplied ("scaled") by numbers called '' scalars''. Scalars are often real numbers, but can ...

of all convergent sequences

\left(x_n\right)

real number In mathematics, a real number is a number that can be used to measure a ''continuous'' one-dimensional quantity such as a distance, duration or temperature. Here, ''continuous'' means that values can have arbitrarily small variations. Every real ...

s or

complex number In mathematics, a complex number is an element of a number system that extends the real numbers with a specific element denoted , called the imaginary unit and satisfying the equation i^= -1; every complex number can be expressed in the form ...

s. When equipped with the uniform norm:

\, x\, _\infty = \sup_n , x_n,

the space

c

becomes a

Banach space In mathematics, more specifically in functional analysis, a Banach space (pronounced ) is a complete normed vector space. Thus, a Banach space is a vector space with a metric that allows the computation of vector length and distance between vector ...

. It is a

closed Closed may refer to: Mathematics * Closure (mathematics), a set, along with operations, for which applying those operations on members always results in a member of the set * Closed set, a set which contains all its limit points * Closed interval, ...

linear subspace In mathematics, and more specifically in linear algebra, a linear subspace, also known as a vector subspaceThe term ''linear subspace'' is sometimes used for referring to flats and affine subspaces. In the case of vector spaces over the reals, li ...

of the

space of bounded sequences 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 n ...

\ell^\infty

, and contains as a closed subspace the Banach space

c_0

of sequences converging to zero. The

dual Dual or Duals may refer to: Paired/two things * Dual (mathematics), a notion of paired concepts that mirror one another ** Dual (category theory), a formalization of mathematical duality *** see more cases in :Duality theories * Dual (grammatical ...

c

is isometrically isomorphic to

\ell^1,

as is that of

c_0.

In particular, neither

c

nor

c_0

is reflexive. In the first case, the isomorphism of

\ell^1

with

c^*

is given as follows. If

\left(x_0, x_1, \ldots\right) \in \ell^1,

then the pairing with an element

\left(y_0, y_1, \ldots\right)

c

is given by

x_0\lim_ y_n + \sum_^\infty x_i y_i.

This is the Riesz representation theorem on the ordinal

\omega.

For

c_0,

the pairing between

\left(x_i\right)

\ell^1

and

\left(y_i\right)

c_0

is given by

\sum_^\infty x_iy_i.

References

* . {{mathanalysis-stub Banach spaces Functional analysis Normed spaces Norms (mathematics)

See also

References