Skip to main content
Disciplines
Automotive Business IT + Informatics Construction + Real Estate Electrical Engineering + Electronics Energy + Sustainability Insurance + Risk Finance + Banking Management + Leadership Marketing + Sales Mechanical Engineering + Materials
Events
DE
EN
Books
Journals
Topic Page
Marketing
Start single access now
Access for companies
EXTENDED SEARCH
Log in
Top
Published in:
Metric and Topological Spaces Read chapter Read first chapter

2018 | OriginalPaper | Chapter

16. Topological Vector Spaces

Author : Vladimir Kadets

Published in: A Course in Functional Analysis and Measure Theory

Publisher: Springer International Publishing

Log in

Activate our intelligent search to find suitable subject content or patents.

search-config
loading …

Abstract

There are natural types of convergence on linear spaces of functions with the feature that the convergence cannot be described as convergence with respect to a norm. These are, for instance, pointwise convergence and convergence in measure. Such types of convergence will, with rare exceptions, be the weak and weak\(^*\) convergence in Banach spaces — the main objects of study in Chapter 17. An adequate language for describing such convergences is that of topological vector spaces. After giving the necessary preliminaries about filters and ultrafilters on topological spaces, in the present chapter we give axiomatics and terminology of topological vector spaces, speak about boundedness, precompactness and compactness in such spaces, discuss the extensions of the Hahn–Banach theorem, and present the elegant Eidelheit’s interpolation theorem with applications to interpolation by infinitely smooth functions and analytic functions.

Dont have a licence yet? Then find out more about our products and how to get one now:

Springer Professional "Wirtschaft+Technik"

Online-Abonnement

Mit Springer Professional "Wirtschaft+Technik" erhalten Sie Zugriff auf:

  • über 102.000 Bücher
  • über 537 Zeitschriften

aus folgenden Fachgebieten:

  • Automobil + Motoren
  • Bauwesen + Immobilien
  • Business IT + Informatik
  • Elektrotechnik + Elektronik
  • Energie + Nachhaltigkeit
  • Finance + Banking
  • Management + Führung
  • Marketing + Vertrieb
  • Maschinenbau + Werkstoffe
  • Versicherung + Risiko

Jetzt Wissensvorsprung sichern!

inform now

Springer Professional "Technik"

Online-Abonnement

Mit Springer Professional "Technik" erhalten Sie Zugriff auf:

  • über 67.000 Bücher
  • über 390 Zeitschriften

aus folgenden Fachgebieten:

  • Automobil + Motoren
  • Bauwesen + Immobilien
  • Business IT + Informatik
  • Elektrotechnik + Elektronik
  • Energie + Nachhaltigkeit
  • Maschinenbau + Werkstoffe




 

Jetzt Wissensvorsprung sichern!

inform now

Springer Professional "Wirtschaft"

Online-Abonnement

Mit Springer Professional "Wirtschaft" erhalten Sie Zugriff auf:

  • über 67.000 Bücher
  • über 340 Zeitschriften

aus folgenden Fachgebieten:

  • Bauwesen + Immobilien
  • Business IT + Informatik
  • Finance + Banking
  • Management + Führung
  • Marketing + Vertrieb
  • Versicherung + Risiko




Jetzt Wissensvorsprung sichern!

inform now
previous chapter Fixed Point Theorems and Applications
next chapter Elements of Duality Theory
Footnotes
1
What a splendid thing is the modern system of notations: \(\mathcal {U}_\mathfrak {A}\) is a familiy of neighborhoods. Each neighborhood is a set of ultrafilters. Each ultrafilter is a family of sets of natural numbers. Thus, with one symbol \(\mathcal {U}_\mathfrak {A}\) we managed to denote a set of sets of sets of sets of natural numbers!
 
2
Here again the already mentioned terminological confusion is widespread. The current term is introduced to generalize the notion of complete metric space. Equally successfully one could have called complete a set whose linear span coincides with the space X (a term used in the theory of linear spaces) or, by analogy with the theory of normed spaces, call a set complete if its linear span is dense in X. We thus obtain identically named notions which however have nothing in common. The relevant meaning must be figured out from the context.
 
3
Often, in textbooks on topological vector spaces, the symbol \(X^*\) is used to denote the set of all linear functionals on X, while the set of continuous linear functionals is denoted by \({X}'\). We will do exactly the opposite, in order to preserve the compatibility with the notations from the theory of normed spaces the reader is already familiar with.
 
4
During the preparation for publication of the second volume of his monograph [40], I. Singer discovered a gap in the original proof of Erdős and Straus. He distributed a letter to other specialists in the theory of bases, asking for an alternative proof of the result. Such proofs were obtained by P. Terenzi and at about the same time by V.I. Gurariĭ, who back then, in 1980, was an active participant in our Kharkiv seminar on the theory of Banach spaces. I have nostalgic memories about those times: in the spring of 1980 I was a third-year student, and this was the first “mature” problem to which I devoted serious thought. The example in Exercise 6 — the fruit of this pondering — was later mentioned by Singer in his monograph. One can imagine how proud I was for discovering this example ...It is amusing that I published this observation only after 10 years and a bit [56].
 
Metadata
Title
Topological Vector Spaces
Author
Vladimir Kadets
Copyright Year
2018
Book
A Course in Functional Analysis and Measure Theory

Print ISBN: 978-3-319-92003-0

Electronic ISBN: 978-3-319-92004-7

Copyright Year: 2018

DOI
https://doi.org/10.1007/978-3-319-92004-7_16

Premium Partner

    Image Credits