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Short description: 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 $ℓ^{\infty}$ under the supremum norm $‖ \cdot ‖_{\infty}$ ^[1]

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

References

↑ ^1.0 ^1.1 ^1.2 Banas, Jozef; Mursaleen, M. (2014), Sequence Spaces and Measures of Noncompactness with Applications to Differential and Integral Equations, Springer, p. 20, ISBN 9788132218869, https://books.google.com/books?id=pWklBAAAQBAJ&pg=PA20 .

v t e Functional analysis (topics)
Topological vector spaces	Asplund Banach (list) Banach lattice Barrelled Bornological Brauner F-space Fréchet (tame) Hilbert (Inner product space Polarization identity) LF-space Locally convex (Seminorms/Minkowski functionals) Mackey Montel Nuclear Normed (norm) Quasinormed Reflexive Riesz Smith Stereotype Strictly convex Webbed Topological tensor product (of Hilbert spaces)
Topologies of function spaces	Dual Dual space (Dual norm) Operator Ultraweak Weak (polar operator) Mackey Strong (polar operator) Ultrastrong Uniform convergence
Linear operators	Adjoint Bilinear (form operator sesquilinear) (Un)Bounded Closed Compact (on Hilbert spaces) (Dis)Continuous Densely defined Fredholm Hilbert–Schmidt Functionals (positive) Normal Nuclear Self-adjoint Strictly singular Trace class Transpose Unitary
Operator theory	Banach algebras C-algebras Spectrum (C-algebra radius) Spectral theory (of ODEs Spectral theorem) Polar decomposition Singular value decomposition
Theorems	Banach–Alaoglu Banach–Mazur Banach–Saks Banach–Schauder (open mapping) Banach–Steinhaus (Uniform boundedness) Bessel's inequality Cauchy–Schwarz inequality Closed graph Closed range Eberlein–Šmulian Freudenthal spectral Gelfand–Mazur Gelfand–Naimark Goldstine Hahn–Banach (hyperplane separation) Kakutani fixed-point Krein–Milman Lomonosov's invariant subspace Mackey–Arens Mazur's lemma M. Riesz extension Riesz representation Parseval's identity Schauder fixed-point
Analysis	Abstract Wiener space Bochner space Differentiation in Fréchet spaces Derivatives (Fréchet Gateaux functional holomorphic) Integrals (Bochner Dunford Gelfand–Pettis regulated Paley–Wiener weak) Functional calculus (Borel continuous holomorphic) Inverse function theorem (Nash–Moser theorem) Measures (Lebesgue Projection-valued Vector) Weakly measurable function
Types of sets	Absolutely convex Absorbing Balanced Bounded Convex Convex cone (subset) Linear cone (subset) Radial Star-shaped Symmetric Zonotope
Subsets / set operations	Algebraic interior (core) Bounding points Convex hull Extreme point Interior Minkowski addition Polar

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BK-space

Examples

See also

References