Dimensions of Metric Spaces

  • Dan A. Simovici
  • Chabane Djeraba
Part of the Advanced Information and Knowledge Processing book series (AI&KP)


Subsets of \({\mathbb R}^n\) may have “intrinsic” dimensions that are much lower than \(n\). Consider, for example, two distinct vectors \(\mathbf {a},\mathbf {b}\in {\mathbb R}^n\) and the line \(L = \{\mathbf {a}+ t \mathbf {b}\,\mid \,t \in {\mathbb R}\}\). Intuitively, \(L\) has the intrinsic dimensionality \(1\); however, \(L\) is embedded in \({\mathbb R}^n\) and from this point of view is an \(n\)-dimensional object. In this chapter we examine formalisms that lead to the definition of this intrinsic dimensionality.


Open Cover Iterative Function System Ultrametric Space Inductive Dimension Outer Measure 
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Copyright information

© Springer-Verlag London 2014

Authors and Affiliations

  1. 1.Department of Computer ScienceUniversity of Massachusetts BostonBostonUSA
  2. 2.University Lille 1Laboratoire d’Informatique Fundamentale de LilleVilleneuve d’AscqFrance

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