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Lecture

Distribution Interpolation Spaces

In course

MATH-502: Distribution and interpolation spaces

The goal of this course is to give an introduction to the theory of distributions and cover the fundamental results of Sobolev spaces including fractional spaces that appear in the interpolation theory. Those notions are central to the study of partial dif ...

Description

This lecture covers distribution interpolation spaces, including the notation used, the regularized flux, the full convergence of functions, extension operators, and the existence of extension operators for open sets of different classes.

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Ontological neighbourhood

Mathematics

Analysis: Functional analysis

Topology: General topology

Related concepts (32)

Hilbert space

In mathematics, Hilbert spaces (named after David Hilbert) allow the methods of linear algebra and calculus to be generalized from (finite-dimensional) Euclidean vector spaces to spaces that may be infinite-dimensional. Hilbert spaces arise naturally and frequently in mathematics and physics, typically as function spaces. Formally, a Hilbert space is a vector space equipped with an inner product that induces a distance function for which the space is a complete metric space.

Compact convergence

In mathematics compact convergence (or uniform convergence on compact sets) is a type of convergence that generalizes the idea of uniform convergence. It is associated with the compact-open topology. Let be a topological space and be a metric space. A sequence of functions is said to converge compactly as to some function if, for every compact set , uniformly on as . This means that for all compact , If and with their usual topologies, with , then converges compactly to the constant function with value 0, but not uniformly.

Open set

In mathematics, an open set is a generalization of an open interval in the real line. In a metric space (a set along with a distance defined between any two points), an open set is a set that, along with every point P, contains all points that are sufficiently near to P (that is, all points whose distance to P is less than some value depending on P). More generally, an open set is a member of a given collection of subsets of a given set, a collection that has the property of containing every union of its members, every finite intersection of its members, the empty set, and the whole set itself.

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In functional analysis, one is interested in extensions of symmetric operators acting on a Hilbert space. Of particular importance is the existence, and sometimes explicit constructions, of self-adjoint extensions. This problem arises, for example, when one needs to specify domains of self-adjointness for formal expressions of observables in quantum mechanics. Other applications of solutions to this problem can be seen in various moment problems. This article discusses a few related problems of this type.

Rigged Hilbert space

In mathematics, a rigged Hilbert space (Gelfand triple, nested Hilbert space, equipped Hilbert space) is a construction designed to link the distribution and square-integrable aspects of functional analysis. Such spaces were introduced to study spectral theory in the broad sense. They bring together the 'bound state' (eigenvector) and 'continuous spectrum', in one place. A function such as is an eigenfunction of the differential operator on the real line R, but isn't square-integrable for the usual Borel measure on R.

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