Densely Defined Operator
Get Densely Defined Operator essential facts below. View Videos or join the Densely Defined Operator discussion. Add Densely Defined Operator to your PopFlock.com topic list for future reference or share this resource on social media.
Densely Defined Operator

In mathematics – specifically, in operator theory – a densely defined operator or partially defined operator is a type of partially defined function. In a topological sense, it is a linear operator that is defined "almost everywhere". Densely defined operators often arise in functional analysis as operations that one would like to apply to a larger class of objects than those for which they a priori "make sense".

Definition

A densely defined linear operator T from one topological vector space, X, to another one, Y, is a linear operator that is defined on a dense linear subspace dom(T) of X and takes values in Y, written T : dom(T) ? X -> Y. Sometimes this is abbreviated as T : X -> Y when the context makes it clear that X might not be the set-theoretic domain of T.

Examples

is a densely defined operator from C0([0, 1]; R) to itself, defined on the dense subspace C1([0, 1]; R). The operator D is an example of an unbounded linear operator, since
has
This unboundedness causes problems if one wishes to somehow continuously extend the differentiation operator D to the whole of C0([0, 1]; R).
  • The Paley-Wiener integral, on the other hand, is an example of a continuous extension of a densely defined operator. In any abstract Wiener space i : H -> E with adjoint j = i* : E* -> H, there is a natural continuous linear operator (in fact it is the inclusion, and is an isometry) from j(E*) to L2(E?R), under which j(f) ? j(E*) ? H goes to the equivalence class [f] of f in L2(E?R). It is not hard to show that j(E*) is dense in H. Since the above inclusion is continuous, there is a unique continuous linear extension I : H -> L2(E?R) of the inclusion j(E*) -> L2(E?R) to the whole of H. This extension is the Paley-Wiener map.

References

  • Renardy, Michael; Rogers, Robert C. (2004). An introduction to partial differential equations. Texts in Applied Mathematics 13 (Second ed.). New York: Springer-Verlag. pp. xiv+434. ISBN 0-387-00444-0. MR 2028503.


  This article uses material from the Wikipedia page available here. It is released under the Creative Commons Attribution-Share-Alike License 3.0.

Densely_defined_operator
 



 



 
Music Scenes