[Math] Good book for measure theory and functional analysis

fa.functional-analysismeasure-theorytextbook-recommendation

I have taken advanced courses both in measure theory and also in functional analysis (Banach and Hilbert spaces, spectral theory of bounded and unbounded operators, etc.)

The connections between the two arises in several theorems:

  1. Riesz theorem showing that under some conditions a continuous functional can be represented as integral with respect to some measure.

  2. Spectral measure and functional calculus for the bounded/unbounded self-adjoint operators.

I have also seen some other results that state that the dual of specific Banach spaces are the same than those of finitely additive measures.

In spite of having advanced course, the connection between measure theory and functional analysis is still really mysterious to me.

I would like to learn more about the connection between the two subjects in a more systematic fashion. I have already seen several related books but the connection is discussed only superficially.

I was wondering if anyone has a suggestion for a rigorous book that focuses specifically on the connection between measure theory and functional analysis.

Best Answer

Very popular are Walter Rudin's books, 1. Functional analysis and 2. Real and Complex Analysis. They cover substantially more than Kolmogorov-Fomin, and from a more modern point of view than Riesz-Nagy.

Another excellent choice is MR0662563
Malliavin, P. Intégration et probabilités. Analyse de Fourier et analyse spectrale, Masson, Paris, 1982. There is an English translation. It has much more measure theory but less complex analysis.

Added. Let me also mention E. Stein and R. Sharkachi, 4 volumes which cover all standard graduate analysis curriculum (Fourier, Complex, Real and Functional analysis). I like and recommend this.

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