space, mathematical, field, functional, analysis, space, consists, infinite, sequences, real, numbers, displaystyle, mathbb, complex, numbers, displaystyle, mathbb, such, thatsup, displaystyle, left, right, finite, such, sequences, forms, normed, space, with, . In the mathematical field of functional analysis the space bs consists of all infinite sequences xi of real numbers R displaystyle mathbb R or complex numbers C displaystyle mathbb C such thatsup n i 1 n x i displaystyle sup n left sum i 1 n x i right is finite The set of such sequences forms a normed space with the vector space operations defined componentwise and the norm given by x b s sup n i 1 n x i displaystyle x bs sup n left sum i 1 n x i right Furthermore with respect to metric induced by this norm bs is complete it is a Banach space The space of all sequences x i displaystyle left x i right such that the series i 1 x i displaystyle sum i 1 infty x i is convergent possibly conditionally is denoted by cs This is a closed vector subspace 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 ℓ displaystyle ell infty via the mappingT x 1 x 2 x 1 x 1 x 2 x 1 x 2 x 3 displaystyle 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 displaystyle T See also editList of Banach spacesReferences editDunford N Schwartz J T 1958 Linear operators Part I Wiley Interscience nbsp This mathematical analysis related article is a stub You can help Wikipedia by expanding it vte Retrieved from https en wikipedia org w index php title Bs space amp oldid 1034623230, wikipedia, wiki, book, books, library,