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Published in:
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2016 | OriginalPaper | Chapter

1. Bases and Basic Sequences

Authors : Fernando Albiac, Nigel J. Kalton

Published in: Topics in Banach Space Theory

Publisher: Springer International Publishing

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Abstract

In this chapter 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 differences and similarities between spaces. The systematic use of basic sequence arguments also turns out to simplify some classical theorems, and we illustrate this with the Eberlein–S̆mulian theorem on weakly compact subsets of a Banach space.

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next chapter The Classical Sequence Spaces
Footnotes
1
On the other hand, the Cantor middle third set , \(\mathcal{C}\), consists of all real numbers x in [0, 1] such that when we write x in ternary form \(x =\sum _{ i=1}^{\infty }a_{i}/3^{i}\), then none of the numbers \(a_{1},a_{2},\ldots\) equals 1 (i.e., either a i  = 0 or a i  = 2). The ternary correspondence from \(\mathcal{C}\) onto \(\Delta\), \(\sum _{i=1}^{\infty }a_{i}/3^{i}\mapsto (a_{1}/2,a_{2}/2,\ldots )\), is a homeomorphism.
 
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Metadata
Title
Bases and Basic Sequences
Authors
Fernando Albiac
Nigel J. Kalton
Copyright Year
2016
Book
Topics in Banach Space Theory

Print ISBN: 978-3-319-31555-3

Electronic ISBN: 978-3-319-31557-7

Copyright Year: 2016

DOI
https://doi.org/10.1007/978-3-319-31557-7_1

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