Inmathematics – specifically, inoperator theory – adensely defined operator orpartially defined operator is a type of partially definedfunction. In atopological sense, it is alinear operator that is defined "almost everywhere". Densely defined operators often arise infunctional analysis as operations that one would like to apply to a larger class of objects than those for which theya priori "make sense".^{[clarification needed]}

Aclosed operator that is used in practice is often densely defined.

Adensely defined linear operator${\displaystyle T}$ from onetopological vector space,${\displaystyle X,}$ to another one,${\displaystyle Y,}$ is a linear operator that is defined on adense linear subspace${\displaystyle \mathrm{dom}(T)}$ of${\displaystyle X}$ and takes values in${\displaystyle Y,}$ written${\displaystyle T:\mathrm{dom}(T)\subseteq X\to Y.}$ Sometimes this is abbreviated as${\displaystyle T:X\to Y}$ when the context makes it clear that${\displaystyle X}$ might not be the set-theoreticdomain of${\displaystyle T.}$

Consider the space${\displaystyle C^0([0,1];\mathbb{R})}$ of allreal-valued,continuous functions defined on the unit interval; let${\displaystyle C^1([0,1];\mathbb{R})}$ denote the subspace consisting of allcontinuously differentiable functions. Equip${\displaystyle C^0([0,1];\mathbb{R})}$ with thesupremum norm${\displaystyle \Vert \cdot {\Vert }_{\mathrm{\infty }}}$; this makes${\displaystyle C^0([0,1];\mathbb{R})}$ into a realBanach space. Thedifferentiation operator${\displaystyle D}$ given by$${\displaystyle (\mathrm{D}u)(x)=u^{\prime }(x)}$$ is a densely defined operator from${\displaystyle C^0([0,1];\mathbb{R})}$ to itself, defined on the dense subspace${\displaystyle C^1([0,1];\mathbb{R}).}$ The operator${\displaystyle \mathrm{D}}$ is an example of anunbounded linear operator, since$${\displaystyle u_n(x)=e^{-nx}\text{ has }\frac{{\Vert \mathrm{D}{u}_n\Vert }_{\mathrm{\infty }}}{{\Vert u_n\Vert }_{\mathrm{\infty }}}=n.}$$This unboundedness causes problems if one wishes to somehow continuously extend the differentiation operator${\displaystyle D}$ to the whole of${\displaystyle C^0([0,1];\mathbb{R}).}$

ThePaley–Wiener integral, on the other hand, is an example of acontinuous extension of a densely defined operator. In anyabstract Wiener space${\displaystyle i:H\to E}$ withadjoint${\displaystyle j:=i^{\ast }:E^{\ast }\to H,}$ there is a naturalcontinuous linear operator (in fact it is the inclusion, and is anisometry) from${\displaystyle j\left(E^{\ast }\right)}$ to${\displaystyle L^2(E,\gamma ;\mathbb{R}),}$ under which${\displaystyle j(f)\in j\left(E^{\ast }\right)\subseteq H}$ goes to theequivalence class${\displaystyle [f]}$ of${\displaystyle f}$ in${\displaystyle L^2(E,\gamma ;\mathbb{R}).}$ It can be shown that${\displaystyle j\left(E^{\ast }\right)}$ is dense in${\displaystyle H.}$ Since the above inclusion is continuous, there is a unique continuous linear extension${\displaystyle I:H\to L^2(E,\gamma ;\mathbb{R})}$ of the inclusion${\displaystyle j\left(E^{\ast }\right)\to L^2(E,\gamma ;\mathbb{R})}$ to the whole of${\displaystyle H.}$ This extension is the Paley–Wiener map.

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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.

• Functional analysis
• Hilbert spaces
• Linear operators
• Operator theory

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