Hilbert Spaces

  • Carlos S. Kubrusly

Abstract

What is missing? The algebraic structure of a normed space allowed us to operate with vectors (addition and scalar multiplication), and its topological structure (the one endowed by the norm) gave us a notion of closeness (by means of the metric generated by the norm), which interacts harmoniously with the algebraic operations. In particular, it provided the notion of the length of a vector. So what is missing if algebra and topology have already been properly laid on the same underlying set? A full geometric structure is still missing. Just algebra and topology are not enough to extend to abstract spaces the geometric concept of relative direction (or angle) between vectors that is familiar in Euclidean geometry. The keyword here is orthogonality, a concept that emerges when we equip a linear space with an inner product. This supplies a tremendously rich structure that leads to remarkable simplifications.

Keywords

Hilbert Space Orthonormal Basis Product Space Selfadjoint Operator Partial Isometry 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Springer Science+Business Media New York 2001

Authors and Affiliations

  • Carlos S. Kubrusly
    • 1
  1. 1.Catholic UniversityRio de JaneiroBrazil

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