BK-space

Sequence space that is Banach

In functional analysis and related areas of mathematics, a BK-space or Banach coordinate space is a sequence space endowed with a suitable norm to turn it into a Banach space. All BK-spaces are normable FK-spaces.^[1]

Examples

The space of convergent sequences $c,$ the space of vanishing sequences $c_{0},$ and the space of bounded sequences $\ell ^{\infty }$ under the supremum norm $\|\cdot \|_{\infty }$ ^[1]

The space of absolutely p-summable sequences $\ell ^{p}$ with $p\geq 1$ and the norm $\|\cdot \|_{p}$ ^[1]

References

^ ^a ^b ^c Banas, Jozef; Mursaleen, M. (2014), Sequence Spaces and Measures of Noncompactness with Applications to Differential and Integral Equations, Springer, p. 20, ISBN 9788132218869.

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)

Banach spaces are:

Barrelled
Complete
F-space
Fréchet
- tame
Locally convex
- Seminorms/Minkowski functionals
Mackey
Metrizable
Normed
- norm
Quasinormed
Stereotype

Function space Topologies

Linear operators

Operator theory

Banach algebras
C*-algebras
Operator space
Spectrum
- C*-algebra
- radius
Spectral theory
- of ODEs
- Spectral theorem
Polar decomposition
Singular value decomposition

Theorems

Analysis

Abstract Wiener space
Banach manifold
- bundle
Bochner space
Convex series
Differentiation in Fréchet spaces
Derivatives
- Fréchet
- Gateaux
- functional
- holomorphic
- quasi
Integrals
Functional calculus
Measures
Weakly / Strongly measurable function

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

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^p
Hilbert H
Morrey–Campanato $L^{\lambda ,p}(\Omega )$
ℓ^p
- $\ell ^{\infty }$
L^p
- $L^{\infty }$
- weighted
Schwartz $S\left(\mathbb {R} ^{n}\right)$
Segal–Bargmann F
Sequence space
Sobolev W^k,p
- Sobolev inequality
Triebel–Lizorkin
Wiener amalgam $W(X,L^{p})$

Applications

v t e Topological vector spaces (TVSs)
Basic concepts	Banach space Completeness Continuous linear operator Linear functional Fréchet space Linear map Locally convex space Metrizability Operator topologies Topological vector space Vector space
Main results	Anderson–Kadec Banach–Alaoglu Closed graph theorem F. Riesz's Hahn–Banach (hyperplane separation Vector-valued Hahn–Banach) Open mapping (Banach–Schauder) Bounded inverse Uniform boundedness (Banach–Steinhaus)
Maps	Bilinear operator form Linear map Almost open Bounded Continuous Closed Compact Densely defined Discontinuous Topological homomorphism Functional Linear Bilinear Sesquilinear Norm Seminorm Sublinear function Transpose
Types of sets	Absolutely convex/disk Absorbing/Radial Affine Balanced/Circled Banach disks Bounding points Bounded Complemented subspace Convex Convex cone (subset) Linear cone (subset) Extreme point Pre-compact/Totally bounded Prevalent/Shy Radial Radially convex/Star-shaped Symmetric
Set operations	Affine hull (Relative) Algebraic interior (core) Convex hull Linear span Minkowski addition Polar (Quasi) Relative interior
Types of TVSs	Asplund B-complete/Ptak Banach (Countably) Barrelled BK-space (Ultra-) Bornological Brauner Complete Convenient (DF)-space Distinguished F-space FK-AK space FK-space Fréchet tame Fréchet Grothendieck Hilbert Infrabarreled Interpolation space K-space LB-space LF-space Locally convex space Mackey (Pseudo)Metrizable Montel Quasibarrelled Quasi-complete Quasinormed (Polynomially Semi-) Reflexive Riesz Schwartz Semi-complete Smith Stereotype (B Strictly Uniformly) convex (Quasi-) Ultrabarrelled Uniformly smooth Webbed With the approximation property
Category

Functional analysis (topics – glossary)

Spaces

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