Skip to main content

General Theory: Topological Aspects

  • 850 Accesses

Part of the Lecture Notes in Mathematics book series (LNM,volume 1986)

Abstract

In Chapter 1, we have analyzed the structure of pip-spaces from the algebraic point of view only, (i.e., the compatibility relation). Here we will discuss primarily the topological structure given by the partial inner product itself. The aim is to tighten the definitions so as to eliminate as many pathologies as possible. The picture that emerges is reassuringly simple: Only two types of pip-spaces seem sufficiently regular to have any practical use, namely lattices of Hilbert spaces (LHS) or Banach spaces (LBS), that we have introduced briefly in the Introduction. Our standard reference on locally convex topological vector spaces (LCS) will be the textbook of Köthe [Köt69]. In addition, for the convenience of the reader, we have collected in Appendix B most of the necessary, but not so familiar, notions needed in the text. Notice that we diverge from [Köt69] for the notation. Namely, for a given dual pair \(<E, F>\), we denote the weak topology on E by σ(E,F), its Mackey topology by t(E,F), and its strong topology by β(E,F) (see Appendix B).

Keywords

  • Hilbert Space
  • Banach Space
  • Dual Pair
  • Strong Topology
  • Pair Versus

These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

This is a preview of subscription content, access via your institution.

Buying options

Chapter
USD   29.95
Price excludes VAT (Canada)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD   39.99
Price excludes VAT (Canada)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD   54.99
Price excludes VAT (Canada)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Learn about institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Jean-Pierre Antoine .

Rights and permissions

Reprints and Permissions

Copyright information

© 2009 Springer-Verlag Berlin Heidelberg

About this chapter

Cite this chapter

Antoine, JP., Trapani, C. (2009). General Theory: Topological Aspects. In: Partial Inner Product Spaces. Lecture Notes in Mathematics(), vol 1986. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-05136-4_2

Download citation