Bs space - Misplaced Pages

In the mathematical field of functional analysis, the space bs consists of all infinite sequences (x_i) of real numbers $\mathbb {R}$ or complex numbers $\mathbb {C}$ such that $\sup _{n}\left|\sum _{i=1}^{n}x_{i}\right|$ is finite. The set of such sequences forms a normed space with the vector space operations defined componentwise, and the norm given by $\|x\|_{bs}=\sup _{n}\left|\sum _{i=1}^{n}x_{i}\right|.$

Furthermore, with respect to metric induced by this norm, bs is complete: it is a Banach space.

The space of all sequences $\left(x_{i}\right)$ such that the series $\sum _{i=1}^{\infty }x_{i}$ 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 $\ell ^{\infty }$ 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

Dunford, N.; Schwartz, J.T. (1958), Linear operators, Part I, Wiley-Interscience.

v t e Banach space topics
Types of Banach spaces	Asplund Banach list Banach lattice Grothendieck Hilbert Inner product space Polarization identity (Polynomially) Reflexive Riesz L-semi-inner product (B Strictly Uniformly) convex Uniformly smooth (Injective Projective) Tensor product (of Hilbert spaces)
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Types of sets	Absolutely convex Absorbing Affine Balanced/Circled Bounded Convex Convex cone (subset) Convex series related ((cs, lcs)-closed, (cs, bcs)-complete, (lower) ideally convex, (Hx), and (Hwx)) Linear cone (subset) Radial Radially convex/Star-shaped Symmetric Zonotope
Subsets / set operations	Affine hull (Relative) Algebraic interior (core) Bounding points Convex hull Extreme point Interior Linear span Minkowski addition Polar (Quasi) Relative interior
Examples	Absolute continuity AC $ba(\Sigma )$ c space Banach coordinate BK Besov $B_{p,q}^{s}(\mathbb {R} )$ Birnbaum–Orlicz Bounded variation BV Bs space Continuous C(K) with K compact Hausdorff Hardy H Hilbert H Morrey–Campanato $L^{\lambda ,p}(\Omega )$ ℓ $\ell ^{\infty }$ L $L^{\infty }$ weighted Schwartz $S\left(\mathbb {R} ^{n}\right)$ Segal–Bargmann F Sequence space Sobolev W Sobolev inequality Triebel–Lizorkin Wiener amalgam $W(X,L^{p})$
Applications	Differential operator Finite element method Mathematical formulation of quantum mechanics Ordinary Differential Equations (ODEs) Validated numerics

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Banach Besov Fréchet Hilbert Hölder Nuclear Orlicz Schwartz Sobolev Topological vector
Properties	Barrelled Complete Dual (Algebraic / Topological) Locally convex Reflexive Separable

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