Abstract
The weak and weak* topologies are introduced for Banach spaces and their duals, and the Banach-Alaoglu theorem is proved. Further topologies are introduced on the spaces of bounded linear operators, and these are placed in the context of the general locally convex vector spaces. Properties of convex bodies are discussed, including the Krein-Milman and Choquet theorems, and several applications of weak* compactness are given. These include Furstenberg’s proof of Weyl’s polynomial equidistribution theorem and elliptic regularity for the Laplace operator at the boundary. The theory of distributions is introduced.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
Author information
Authors and Affiliations
Rights and permissions
Copyright information
© 2017 Springer International Publishing AG
About this chapter
Cite this chapter
Einsiedler, M., Ward, T. (2017). Locally Convex Vector Spaces. In: Functional Analysis, Spectral Theory, and Applications. Graduate Texts in Mathematics, vol 276. Springer, Cham. https://doi.org/10.1007/978-3-319-58540-6_8
Download citation
DOI: https://doi.org/10.1007/978-3-319-58540-6_8
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-319-58539-0
Online ISBN: 978-3-319-58540-6
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)