Frame Systems in Non-Locally Convex Banach Spaces

Abstract

In this paper, we define atomic decompositions in a non- locally convex Banach space lp(0 < p ≤ 1) and discuss its existence through examples. Also, a sufficient condition for its existence is given and it is observed that if a p-Banach space has an atomic decomposition, then the space is isomorphic to its associated p-Banach sequence space. Further, necessary and sufficient conditions for an atomic
decomposition in a p-Banach space is given. Finally, we define shrinking atomic decomposition and gave a necessary and sufficient condition for it.

Key words and phrases. p-Banach Space, p-normed linear space, Hahn Banach Theorem.

2010 Mathematics Subject Classification. 42C15; 42C30