Densely defined operator

Examples

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 ${\displaystyle \Vert \cdot {\Vert }_{\mathrm{\infty }}}$; this makes ${\displaystyle C^0([0,1];\mathbb{R})}$ into a real Banach space. The differentiation operator ${\displaystyle D}$ given by $${\displaystyle (\mathrm{D}u)(x)=u^{\prime }(x)}$$ 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 an unbounded 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 ${\displaystyle C^0([0,1];\mathbb{R}).}$

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