Lecture

Calculus of Variations: Radon-Nikodym Theorem

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In course

MATH-604: Some Aspects of Calculus of Variations

The goal of this course is to present an overview on the solvability and the regularity of relevant models of physical, technological and economical systems, which may be formulated as minimization problems of suitable integral functionals. ...

Description

This lecture covers the Radon-Nikodym theorem, which deals with absolutely continuous and penitive functions. It explains the disintegration of measures, the Riesz-Karkov Representation Theorem, and the uniqueness of signed Borel measures. The lecture also discusses the extension of operators, bounded linear transformations, and the essential uniqueness of measures.

Official source

Ontological neighbourhood

Mathematics

Analysis: Functional analysis

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Unbounded operator

In mathematics, more specifically functional analysis and operator theory, the notion of unbounded operator provides an abstract framework for dealing with differential operators, unbounded observables in quantum mechanics, and other cases. The term "unbounded operator" can be misleading, since "unbounded" should sometimes be understood as "not necessarily bounded"; "operator" should be understood as "linear operator" (as in the case of "bounded operator"); the domain of the operator is a linear subspace, not necessarily the whole space; this linear subspace is not necessarily closed; often (but not always) it is assumed to be dense; in the special case of a bounded operator, still, the domain is usually assumed to be the whole space.

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