Vector Spaces

  • Igor R. Shafarevich
  • Alexey O. Remizov


In the third chapter we move to a more abstract and general level. The notions of vector space, subspace, dimension, basis, linear transformations, isomorphism, etc. are introduced and discussed. At the end of this chapter, the notions of dual vector space and forms and polynomials in vectors are considered. Most of the abstract concepts are illustrated with various examples and applications. For instance, the notion of a dual space is accompanied with the idea of “generalized functions” (distributions), and the notions of forms and polynomials in vectors are accompanied with Euler’s identity for homogeneous polynomials.

Copyright information

© Springer-Verlag Berlin Heidelberg 2012

Authors and Affiliations

  • Igor R. Shafarevich
    • 1
  • Alexey O. Remizov
    • 2
  1. 1.Steklov Mathematical InstituteRussian Academy of SciencesMoscowRussia
  2. 2.CMAPÉcole Polytechnique CNRSPalaiseau CedexFrance

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