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Adjoint space

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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.

References

[1] D.A. Raikov, "Vector spaces" , Noordhoff (1965) (Translated from Russian)

Comments

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.

References

[a1] H.H. Schaefer, "Topological vector spaces" , Macmillan (1966)
How to Cite This Entry:
Adjoint space. Encyclopedia of Mathematics. URL: http://www.encyclopediaofmath.org/index.php?title=Adjoint_space&oldid=28995