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".[clarification needed]

A closed operator that is used in practice is often densely defined.

Definition

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

Examples

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Consider the space   of all real-valued, continuous functions defined on the unit interval; let   denote the subspace consisting of all continuously differentiable functions. Equip   with the supremum norm  ; this makes   into a real Banach space. The differentiation operator   given by   is a densely defined operator from   to itself, defined on the dense subspace   The operator   is an example of an unbounded linear operator, since   This unboundedness causes problems if one wishes to somehow continuously extend the differentiation operator   to the whole of  

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   with adjoint   there is a natural continuous linear operator (in fact it is the inclusion, and is an isometry) from   to   under which   goes to the equivalence class   of   in   It can be shown that   is dense in   Since the above inclusion is continuous, there is a unique continuous linear extension   of the inclusion   to the whole of   This extension is the Paley–Wiener map.

See also

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References

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  • 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.