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{{short description|Function that is defined almost everywhere (mathematics)}}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 operatorname{dom}(T) of X and takes values in Y, written T : operatorname{dom}(T) subseteq X to Y. Sometimes this is abbreviated as T : X to Y when the context makes it clear that X might not be the set-theoretic domain of T.

Examples

Consider the space C^0([0, 1]; R) of all real-valued, continuous functions defined on the unit interval; let C^1([0, 1]; R) denote the subspace consisting of all continuously differentiable functions. Equip C^0([0, 1]; R) with the supremum norm |,cdot,|_infty; this makes C^0([0, 1]; R) into a real Banach space. The differentiation operator D given by (mathrm{D} u)(x) = u'(x) is a densely defined operator from C^0([0, 1]; R) to itself, defined on the dense subspace C^1([0, 1]; R). The operator mathrm{D} is an example of an unbounded linear operator, since u_n (x) = e^{- n x} quad text{ has } quad frac{left|mathrm{D} u_nright|_{infty}}{left|u_nright|_infty} = n.This unboundedness causes problems if one wishes to somehow continuously extend the differentiation operator D to the whole of C^0([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 to E with adjoint j := i^* : E^* to H, there is a natural continuous linear operator (in fact it is the inclusion, and is an isometry) from jleft(E^*right) to L^2(E, gamma; R), under which j(f) in jleft(E^*right) subseteq H goes to the equivalence class [f] of f in L^2(E, gamma; R). It can be shown that jleft(E^*right) is dense in H. Since the above inclusion is continuous, there is a unique continuous linear extension I : H to L^2(E, gamma; R) of the inclusion jleft(E^*right) to L^2(E, gamma; R) to the whole of H. This extension is the Paley–Wiener map.

See also

• {{annotated link|Blumberg theorem}}
• {{annotated link|Closed graph theorem (functional analysis)}}
• {{annotated link|Linear extension (linear algebra)}}
• {{annotated link|Partial function}}

References

{{reflist}}

• BOOK, Renardy, Michael, Rogers, Robert C., An introduction to partial differential equations, Texts in Applied Mathematics 13, Second, Springer-Verlag, New York, 2004, xiv+434, 0-387-00444-0, 2028503

, {{Hilbert space}}{{Banach spaces}}{{Functional analysis}}