Now we reach the centerpiece of this volume, which is the theory of bases in Banach spaces. Since every Banach space is a vector space, it has a basis in the ordinary vector space sense, i.e., a set that spans and is linearly independent.However, this definition of basis restricts us to using only finite linear combinationsof vectors, while in any normed space it makes sense to deal with infinite series. Restricting to finite linear combinations when working in an infinite-dimensional space is simply too restrictive for most purposes. Moreover,the proof that a vector space basis exists is nonconstructive in general, as it relies on the Axiom of Choice. Hence we need a new notion of basis that is appropriate for infinite-dimensional Banach spaces, and that is the maintopic of this chapter.

Unable to display preview. Download preview PDF.

Authors and Affiliations

1. School of Mathematics, Georgia Institute of Technology, Atlanta, GA, 30332-0160, USA

   Christopher Heil

Corresponding author

Correspondence toChristopher Heil.

Reprints and permissions

© 2011 Birkhäuser Boston

Policies and ethics