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FK-AK space

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Infunctional analysis and related areas ofmathematics, anFK-AK space orFK-space with theAK property is anFK-space which contains thespace of finite sequences and has aSchauder basis.^[1]

Examples and non-examples

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$c_{0}$ thespace of convergent sequences with thesupremum norm has the AK property.
$\ell ^{p}$ ( $1\leq p<\infty$ ) theabsolutely p-summable sequences with the $\|\cdot \|_{p}$ norm have the AK property.
$\ell ^{\infty }$ with the supremum norm doesnot have the AK property.

Properties

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An FK-AK space $E {\displaystyle E}$ has the property $E'\simeq E^{\beta }$ that is thecontinuous dual of $E {\displaystyle E}$ islinear isomorphic to thebeta dual of $E . {\displaystyle E.}$

FK-AK spaces areseparable spaces.

References

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^Das, Gokulananda; Nanda, Sudarsan (2022).Banach limit and applications (1st ed.). Boca Raton: CRC Press.ISBN 978-1-000-46757-4.

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