K-space (functional analysis)

In mathematics, more specifically in functional analysis, a K-space is an F-space V {\displaystyle V} such that every extension of F-spaces (or twisted sum) of the form  0 → R → X → V → 0. {\displaystyle 0\rightarrow \mathbb {R} \rightarrow X\rightarrow V\rightarrow 0.\,\\!} is equivalent to the trivial one[1] 0 → R → R × V → V → 0. {\displaystyle 0\rightarrow \mathbb {R} \rightarrow \mathbb {R} \times V\rightarrow V\rightarrow 0.\,\\!} where  R {\displaystyle \mathbb {R} } is the real line.

## Examples

[edit]

The  ℓ p {\displaystyle \ell ^{p}} spaces for  0 < p < 1 {\displaystyle 0<p<1} are K-spaces,[1] as are all finite dimensional Banach spaces.

N. J. Kalton and N. P. Roberts proved that the Banach space  ℓ 1 {\displaystyle \ell ^{1}} is not a K-space.[1]

## See also

[edit]

* Compactly generated space – Property of topological spaces
* Gelfand–Shilov space

## References

[edit]

1. ^ a b c Kalton, N. J.; Peck, N. T.; Roberts, James W. An F-space sampler. London Mathematical Society Lecture Note Series, 89. Cambridge University Press, Cambridge, 1984. xii+240 pp. ISBN 0-521-27585-7

* v
* t
* e

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

Theorems|

* Hahn–Banach
* Riesz representation
* Closed graph
* Uniform boundedness principle
* Kakutani fixed-point
* Krein–Milman
* Min–max
* Gelfand–Naimark
* Banach–Alaoglu

Operators|

* Adjoint
* Bounded
* Compact
* Hilbert–Schmidt
* Normal
* Nuclear
* Trace class
* Transpose
* Unbounded
* Unitary

Algebras|

* Banach algebra
* C*-algebra
* Spectrum of a C*-algebra
* Operator algebra
* Group algebra of a locally compact group
* Von Neumann algebra

Open problems|

* Invariant subspace problem
* Mahler's conjecture

Applications|

* Hardy space
* Spectral theory of ordinary differential equations
* Heat kernel
* Index theorem
* Calculus of variations
* Functional calculus
* Integral linear operator
* Jones polynomial
* Topological quantum field theory
* Noncommutative geometry
* Riemann hypothesis
* Distribution (or Generalized functions)

Advanced topics|

* Approximation property
* Balanced set
* Choquet theory
* Weak topology
* Banach–Mazur distance
* Tomita–Takesaki theory

* Category

* 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

*[v]: View this template
*[t]: Discuss this template
*[e]: Edit this template