Normed spaces

  • Charalambos D. Aliprantis
  • Kim C. Border
Part of the Studies in Economic Theory book series (ECON.THEORY, volume 4)


This chapter studies some of the special properties of normed spaces. All finite dimensional spaces have a natural norm, the Euclidean norm. On a finite dimensional vector space, the Hausdorif linear topology the norm generates is unique (Theorem 4.61). The Euclidean norm makes ℝn into a complete metric space. A normed space that is complete in the metric induced by its norm is called a Banach space. Here is an overview of some of the more salient results in this chapter.


Banach Space Normed Space Weak Topology Dual Pair Norm Topology 
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  1. 1.
    The natural duality of a normed space with its norm dual is not always the most useful pairing. Two important examples are the normed spaces Bb(Х) of bounded Borel measurable functions on a metrizable space, and the space L (μ) of μ-integrable functions. (Both include ℓ as a special case.) The dual of Bb is the space of bounded charges, but the pairing (Bb,ca) of Bb with finite measures is more common. See Section 11.6 for a discussion of this pair. Similarly, the dual of L is larger than L1, but the pairing (L, L1) is more useful. This can be confusing at times. For instance, the τ(ℓ, ℓ1) Mackey topology for the dual pair (ℓ, ℓ1) is not the norm topology on ℓ: it is weaker. In this chapter at least, we do not deal with other pairings. But when it comes to applying these theorems, make sure you know your dual.Google Scholar
  2. 2.
    It is also customary to denote the value T( x ) by Tx.Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1994

Authors and Affiliations

  • Charalambos D. Aliprantis
    • 1
  • Kim C. Border
    • 2
  1. 1.Department of MathematicsIndiana University-Purdue University Indianapolis (IUPUI)IndianapolisUSA
  2. 2.Division of the Humanities and Social SciencesCALTECHPasadenaUSA

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