IDEAS home Printed from https://ideas.repec.org/h/spr/sprchp/978-0-8176-8232-3_9.html

Normed and Function Spaces

In: Basic Real Analysis

Author

Listed:
  • Houshang H. Sohrab

    (Towson University, Mathematics Department)

Abstract

Metric spaces were defined and studied in Chapter 5. Despite their importance, general metric spaces do not share the algebraic properties of the fields ℝ and ℂ, because a metric space need not be an algebra or even a vector space. Any (nonempty) subset of a metric space is again a metric space with the metric it borrows from the ambient space. Thus, curves and surfaces in the Euclidean space ℝ3 are metric spaces but are almost never vector subspaces of ℝ3. Even straight lines and planes are not vector subspaces unless they pass through the origin. There is an important class of metric spaces, however, that is a natural framework for the extension of topological as well as algebraic properties of ℝ and ℂ : It is the class of Normed Spaces, which we now define. Throughout this chapter, $$ \mathbb{F} $$ will stand for either ℝ or ℂ and X, Y, Z, etc. will denote vector spaces over $$ \mathbb{F} $$ .

Suggested Citation

  • Houshang H. Sohrab, 2003. "Normed and Function Spaces," Springer Books, in: Basic Real Analysis, chapter 9, pages 351-393, Springer.
    • Handle: RePEc:spr:sprchp:978-0-8176-8232-3_9
    DOI: 10.1007/978-0-8176-8232-3_9
    as

    Download full text from publisher

    To our knowledge, this item is not available for download. To find whether it is available, there are three options:
    1. Check below whether another version of this item is available online.
    2. Check on the provider's web page whether it is in fact available.
    3. Perform a
    for a similarly titled item that would be available.

    More about this item

    Keywords

    ;
    ;
    ;
    ;
    ;

    Statistics

    Access and download statistics

    Corrections

    All material on this site has been provided by the respective publishers and authors. You can help correct errors and omissions. When requesting a correction, please mention this item's handle: RePEc:spr:sprchp:978-0-8176-8232-3_9. See general information about how to correct material in RePEc.

    If you have authored this item and are not yet registered with RePEc, we encourage you to do it here. This allows to link your profile to this item. It also allows you to accept potential citations to this item that we are uncertain about.

    We have no bibliographic references for this item. You can help adding them by using this form .

    If you know of missing items citing this one, you can help us creating those links by adding the relevant references in the same way as above, for each refering item. If you are a registered author of this item, you may also want to check the "citations" tab in your RePEc Author Service profile, as there may be some citations waiting for confirmation.

    For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: Sonal Shukla or Springer Nature Abstracting and Indexing (email available below). General contact details of provider: http://www.springer.com .

    Please note that corrections may take a couple of weeks to filter through the various RePEc services.

    IDEAS is a RePEc service. RePEc uses bibliographic data supplied by the respective publishers.