Bases in Banach Spaces

Kumar, Adarsh and Paul, Tanmoy (2018) Bases in Banach Spaces. Masters thesis, Indian Institute of Technology Hyderabad.

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Abstract

Countable basis is similar to the usual (Hamel) basis of a vector space; the di�erence is that Hamel bases use linear combinations that are �nite sums, while for Schauder bases they may be in�nite sums. This makes Schauder bases more suitable for the analysis of in�nite-dimensional Banach spaces.It is therefore very natural to look for a concept to extend the notion of a basis of a finite dimensional vector space.we are going to introduce the fundamental notion of a Schauder basis of a Banach space and the corresponding notion of a basic sequence. One of the key ideas in the isomorphic theory of Banach spaces is to use the properties of bases and basic sequences as a tool to understanding the di�erences and similarities between spaces. The systematic use of basic sequence arguments also turns out to simplify some classical theorems.

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