Densely defined operator

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

Consider the space ${\displaystyle C^{0}([0,1];\mathbb {R} )}$ of all real-valued, continuous functions defined on the unit interval; let ${\displaystyle C^{1}([0,1];\mathbb {R} )}$ denote the subspace consisting of all continuously differentiable functions. Equip ${\displaystyle C^{0}([0,1];\mathbb {R} )}$ with the supremum norm ${\displaystyle \|\,\cdot \,\|_{\infty }}$; this makes ${\displaystyle C^{0}([0,1];\mathbb {R} )}$ into a real Banach space. The differentiation operator ${\displaystyle D}$ given by

$${\displaystyle (\mathrm {D} u)(x)=u'(x)}$$

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 an unbounded linear operator, since

$${\displaystyle u_{n}(x)=e^{-nx}\quad {\text{ has }}\quad {\frac {\left\|\mathrm {D} u_{n}\right\|_{\infty }}{\left\|u_{n}\right\|_{\infty }}}=n.}$$

This unboundedness causes problems if one wishes to somehow continuously extend the differentiation operator ${\displaystyle D}$ to the whole of ${\displaystyle C^{0}([0,1];\mathbb {R} ).}$

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 ${\displaystyle i:H\to E}$ with adjoint ${\displaystyle j:=i^{*}:E^{*}\to H,}$ there is a natural continuous linear operator (in fact it is the inclusion, and is an isometry) from ${\displaystyle j\left(E^{*}\right)}$ to ${\displaystyle L^{2}(E,\gamma$ ;\mathbb {R} ),} under which ${\displaystyle j(f)\in j\left(E^{*}\right)\subseteq H}$ goes to the equivalence class ${\displaystyle [f]}$ of ${\displaystyle f}$ in ${\displaystyle L^{2}(E,\gamma$ ;\mathbb {R} ).} It can be shown that ${\displaystyle j\left(E^{*}\right)}$ 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 ${\displaystyle j\left(E^{*}\right)\to L^{2}(E,\gamma$ ;\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.