Adjoint Space

of a topological vector space $E$

The vector space $E^{*}$ consisting of continuous linear functions on $E$. If $E$ is a locally convex space, then the functionals $f\in E^{*}$ separate the points of $E$ (the Hahn–Banach theorem). If $E$ is a normed space, then $E^{*}$ is a Banach space with respect to the norm \begin{equation*} \|f\| = \sup\limits_{x\ne0}\frac{|f(x)|}{\|x\|}. \end{equation*} There are two (usually different) natural topologies on $E^{*}$ which are often used: the strong topology determined by this norm and the weak-$*$-topology.

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Instead of the term adjoint space one more often uses the term dual space. The weak-$*$-topology on $E^{*}$ is the weakest topology on $E^{*}$ for which all the evaluation mappings $f\mapsto f(x)$, $f\in E^{*}$, $x\in E$, are continuous.

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