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.

Examples and non-examples

• $c_0$ the space of convergent sequences with the supremum norm has the AK property.
• ${\ell }^p$ ($1\le p<\mathrm{\infty }$) the absolutely p-summable sequences with the $\Vert \cdot {\Vert }_p$ norm have the AK property.
• ${\ell }^{\mathrm{\infty }}$ with the supremum norm does not have the AK property.

Properties

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

FK-AK spaces are separable spaces.

See also

• BK-space – Sequence space that is Banach
• FK-space
• Normed space
• Sequence space – Vector space of infinite sequences