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Functional Analysis: Entering Hilbert Space

Vagn Lundsgaard Hansen
World Scientific
Publication Date: 
Number of Pages: 
[Reviewed by
Kenneth A. Ross
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This is a well-written student-friendly basic introduction to functional analysis and Hilbert space, culminating in the spectral theorem for self-adjoint linear operators on separable Hilbert space. The detailed proofs are easy to read, and the exercises are reasonable. Occasionally the presentation is too slick. The author avoids weak and strong topologies, even in Hilbert space, and this makes for a cumbersome unintuitive proof that every weakly convergent sequence in a Hilbert space is bounded.

My one complaint is that the author introduces the \(L^p\)-spaces on \(\mathbb{R}\) and proves the standard inequalities in a mathematically-correct way that is much too sophisticated compared with the rest of the book. In a remark, he states that “it’s not necessary to know the exact nature of the elements in \(L^p\). It is sufficient to work with them as limits of Cauchy sequences of continuous functions, with compact support, with respect to the \(p\)-norm.” This is technically correct, but from here on the author uses function notation and later on defines a set to be measurable if its characteristic function is of class \(L^1\). It is true, and not difficult with a little Lebesgue theory, to prove that \(L^p\) as defined above consists of (measurable) functions on \(\mathbb{R}\).

Kenneth A. Ross ( taught at the University of Oregon from 1965 to 2000. He was President of the MAA during 1995-1996. Before that he served as AMS Associate Secretary, MAA Secretary, and MAA Associate Secretary. His research area of interest was commutative harmonic analysis, especially where it has a probabilistic flavor. He is the author of the book Elementary Analysis: The Theory of Calculus (1980, now in 14th printing), co-author of Discrete Mathematics (with Charles R.B. Wright, 2003, fifth edition), and, as Ken Ross, the author of A Mathematician at the Ballpark: Odds and Probabilities for Baseball Fans (2004).

  • Basic Elements of Metric Topology
  • New Types of Function Spaces
  • Theory of Hilbert Spaces
  • Operators on Hilbert Spaces
  • Spectral Theory
  • Exercises and Applications