 Densely Defined Operator
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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

$(\mathrm {D} u)(x)=u'(x)\,$ 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
$u_{n}(x)=e^{-nx}\,$ has
${\frac {\|\mathrm {D} u_{n}\|_{\infty }}{\|u_{n}\|_{\infty }}}=n.$ 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.