In the

mathematical Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...

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, norm, topology, etc.) and the linear functions defined o ...

, the space bs consists of all infinite

sequences In mathematics, a sequence is an enumerated collection of objects in which repetitions are allowed and order matters. Like a set, it contains members (also called ''elements'', or ''terms''). The number of elements (possibly infinite) is called t ...

(''x''_''i'') of real numbers

\R

complex numbers 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 ...

\Complex

such that

\sup_n \left, \sum_^n x_i\

is finite. The set of such sequences forms a normed space with 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 ...

operations defined componentwise, and the norm given by

\, x\, _ = \sup_n \left, \sum_^n x_i\.

Furthermore, with respect to

metric Metric or metrical may refer to: * Metric system, an internationally adopted decimal system of measurement * An adjective indicating relation to measurement in general, or a noun describing a specific type of measurement Mathematics In mathem ...

induced by this norm, ''bs'' is complete: it is a Banach space. The space of all sequences

\left(x_i\right)

such that the

series Series may refer to: People with the name * Caroline Series (born 1951), English mathematician, daughter of George Series * George Series (1920–1995), English physicist Arts, entertainment, and media Music * Series, the ordered sets used in ...

\sum_^\infty x_i

is convergent (possibly conditionally) is denoted by ''cs''. This is a closed

vector 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, l ...

of ''bs'', and so is also a Banach space with the same norm. The space ''bs'' is isometrically isomorphic to the Space of bounded sequences

\ell^

via the mapping

T(x_1, x_2, \ldots) = (x_1, x_1+x_2, x_1+x_2+x_3, \ldots).

Furthermore, the space of convergent sequences ''c'' is the image of ''cs'' under

T.

References

* . Banach spaces Functional analysis {{mathanalysis-stub

See also

References