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In mathematics – specifically, in
operator theory In mathematics, operator theory is the study of linear operators on function spaces, beginning with differential operators and integral operators. The operators may be presented abstractly by their characteristics, such as bounded linear oper ...
– a densely defined operator or partially defined operator is a type of partially defined function. In a
topological In mathematics, topology (from the Greek words , and ) is concerned with the properties of a geometric object that are preserved under continuous deformations, such as stretching, twisting, crumpling, and bending; that is, without closing ...
sense, it is a
linear operator In mathematics, and more specifically in linear algebra, a linear map (also called a linear mapping, linear transformation, vector space homomorphism, or in some contexts linear function) is a Map (mathematics), mapping V \to W between two vect ...
that is defined "almost everywhere". Densely defined operators often arise in
functional analysis Functional analysis is a branch of mathematical analysis, the core of which is formed by the study of vector spaces endowed with some kind of limit-related structure (e.g. inner product, norm, topology, etc.) and the linear functions defined ...
as operations that one would like to apply to a larger class of objects than those for which they ''
a priori ("from the earlier") and ("from the later") are Latin phrases used in philosophy to distinguish types of knowledge, justification, or argument by their reliance on empirical evidence or experience. knowledge is independent from current ex ...
'' "make sense".


Definition

A densely defined linear operator T from one
topological vector space In mathematics, a topological vector space (also called a linear topological space and commonly abbreviated TVS or t.v.s.) is one of the basic structures investigated in functional analysis. A topological vector space is a vector space that is al ...
, X, to another one, Y, is a linear operator that is defined on a
dense Density (volumetric mass density or specific mass) is the substance's mass per unit of volume. The symbol most often used for density is ''ρ'' (the lower case Greek letter rho), although the Latin letter ''D'' can also be used. Mathematically ...
linear subspace \operatorname(T) of X and takes values in Y, written T : \operatorname(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 Domain may refer to: Mathematics *Domain of a function, the set of input values for which the (total) function is defined ** Domain of definition of a partial function **Natural domain of a partial function **Domain of holomorphy of a function *Do ...
of T.


Examples

Consider the space C^0(
, 1 The comma is a punctuation mark that appears in several variants in different languages. It has the same shape as an apostrophe or single closing quotation mark () in many typefaces, but it differs from them in being placed on the baseline o ...
\R) of all
real-valued In mathematics, value may refer to several, strongly related notions. In general, a mathematical value may be any definite mathematical object. In elementary mathematics, this is most often a number – for example, a real number such as or an ...
, continuous functions defined on the unit interval; let C^1(
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\R) denote the subspace consisting of all continuously differentiable functions. Equip C^0(
, 1 The comma is a punctuation mark that appears in several variants in different languages. It has the same shape as an apostrophe or single closing quotation mark () in many typefaces, but it differs from them in being placed on the baseline o ...
\R) with the
supremum norm In mathematical analysis, the uniform norm (or ) assigns to real- or complex-valued bounded functions defined on a set the non-negative number :\, f\, _\infty = \, f\, _ = \sup\left\. This norm is also called the , the , the , or, when ...
\, \,\cdot\,\, _\infty; this makes C^0(
, 1 The comma is a punctuation mark that appears in several variants in different languages. It has the same shape as an apostrophe or single closing quotation mark () in many typefaces, but it differs from them in being placed on the baseline o ...
\R) into a real
Banach space In mathematics, more specifically in functional analysis, a Banach space (pronounced ) is a complete normed vector space. Thus, a Banach space is a vector space with a metric that allows the computation of vector length and distance between ve ...
. The
differentiation operator In mathematics, a differential operator is an operator defined as a function of the differentiation operator. It is helpful, as a matter of notation first, to consider differentiation as an abstract operation that accepts a function and retur ...
D given by (\mathrm u)(x) = u'(x) is a densely defined operator from C^0(
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\R) to itself, defined on the dense subspace C^1(
, 1 The comma is a punctuation mark that appears in several variants in different languages. It has the same shape as an apostrophe or single closing quotation mark () in many typefaces, but it differs from them in being placed on the baseline o ...
\R). The operator \mathrm is an example of an unbounded linear operator, since u_n (x) = e^ \quad \text \quad \frac = n. This unboundedness causes problems if one wishes to somehow continuously extend the differentiation operator D to the whole of C^0(
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\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 In mathematics, the term ''adjoint'' applies in several situations. Several of these share a similar formalism: if ''A'' is adjoint to ''B'', then there is typically some formula of the type :(''Ax'', ''y'') = (''x'', ''By''). Specifically, adjoin ...
j := i^* : E^* \to H, there is a natural
continuous linear operator In functional analysis and related areas of mathematics, a continuous linear operator or continuous linear mapping is a continuous linear transformation between topological vector spaces. An operator between two normed spaces is a bounded line ...
(in fact it is the inclusion, and is an
isometry In mathematics, an isometry (or congruence, or congruent transformation) is a distance-preserving transformation between metric spaces, usually assumed to be bijective. The word isometry is derived from the Ancient Greek: ἴσος ''isos'' mea ...
) from j\left(E^*\right) to L^2(E, \gamma; \R), under which j(f) \in j\left(E^*\right) \subseteq H goes to the
equivalence class In mathematics, when the elements of some set S have a notion of equivalence (formalized as an equivalence relation), then one may naturally split the set S into equivalence classes. These equivalence classes are constructed so that elements ...
/math> of f in L^2(E, \gamma; \R). It can be shown that j\left(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 j\left(E^*\right) \to L^2(E, \gamma; \R) to the whole of H. This extension is the Paley–Wiener map.


See also

* * *


References

* {{DEFAULTSORT:Densely-Defined Operator Functional analysis Hilbert space Linear operators Operator theory