FK-AK space

In functional analysis and related areas of mathematics, an FK-AK space or FK-space with the AK property is an FK-space which contains the space of finite sequences and has a Schauder basis. ^[1]

Examples and non-examples

• ${\displaystyle c_{0}}$ the space of convergent sequences with the supremum norm has the AK property.
• ${\displaystyle \ell ^{p}}$ (${\displaystyle 1\leq p<\infty }$) the absolutely p-summable sequences with the ${\displaystyle \|\cdot \|_{p}}$ norm have the AK property.
• ${\displaystyle \ell ^{\infty }}$ with the supremum norm does not have the AK property.

Properties

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

FK-AK spaces are separable spaces.

See also

• BK-space  – Sequence space that is Banach
• FK-space  – Sequence space that is Fréchet
• Normed space  – Vector space on which a distance is definedPages displaying short descriptions of redirect targets
• Sequence space  – Vector space of infinite sequences

References

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