Annals of Functional Analysis

Two applications of nets

Ralf Beckmann and Anton Deitmar

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Two applications of nets are given. The first is an extension of the Bochner integral to arbitrary locally convex spaces, leading to an integration theory of more general vector valued functions then in the classical approach by Gelfand and Pettis. The second application starts with the observation that an operator on a Hilbert space is trace class if and only if the net of "principal trace minors" converges. The notion of a "determinant class operator" then is defined as one for which the net of determinantal principal minors converges. It is shown that for a normal operator $A$ this condition coincides with $1-A$ being trace class.

Article information

Ann. Funct. Anal., Volume 6, Number 3 (2015), 176-190.

First available in Project Euclid: 17 April 2015

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 46E40: Spaces of vector- and operator-valued functions
Secondary: 28B05: Vector-valued set functions, measures and integrals [See also 46G10] 46G10: Vector-valued measures and integration [See also 28Bxx, 46B22] 47A05: General (adjoints, conjugates, products, inverses, domains, ranges, etc.) 47B10: Operators belonging to operator ideals (nuclear, p-summing, in the Schatten-von Neumann classes, etc.) [See also 47L20]

Bochner-integral net trace determinant


Beckmann, Ralf; Deitmar, Anton. Two applications of nets. Ann. Funct. Anal. 6 (2015), no. 3, 176--190. doi:10.15352/afa/06-3-15.

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