On some locally convex FK spaces
Paolo Leonetti, Cihan Orhan

TL;DR
This paper characterizes conditions under which certain vector spaces of real sequences are locally convex FK spaces, focusing on the inclusion relations with $c_{00}( ext{I})$ and $ ext{ell}_ ext{infinity}( ext{I})$, and the properties of the ideal $ ext{I}$.
Contribution
It provides necessary and sufficient conditions for vector spaces of sequences to be locally convex FK spaces, especially in relation to ideals on $ ext{omega}$ and inclusion properties.
Findings
Characterization of when $V$ is a locally convex FK space based on ideal properties.
Equivalence of FK space condition with the existence of an infinite set $S$ with specific ideal properties.
Conditions linking sequence space inclusions with the structure of the ideal $ ext{I}$.
Abstract
We provide necessary and/or sufficient conditions on vector spaces of real sequences to be a Fr\'{e}chet space such that each coordinate map is continuous, that is, to be a locally convex FK space. In particular, we show that if for some ideal on , then is a locally convex FK space if and only if there exists an infinite set for which every infinite subset does not belong to .
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Taxonomy
TopicsRings, Modules, and Algebras · Advanced Banach Space Theory · Advanced Topics in Algebra
