I've been reading Ballentine, Chapter 1. Have I got this the right way around? Taking our inner product to be linear in its second argument and conjugate linear in its first, the (continuous?) conjugate space of a Hilbert space [itex]\cal{H}[/itex] is the following set of linear functionals, each identified with an element of [itex]\cal{H}[/itex] via the isomorphism defined by the inner product:(adsbygoogle = window.adsbygoogle || []).push({});

[tex]\cal{H}^{\times}=\left \{ F_\alpha \text{ continuous } \; | \; F_\alpha(\beta) \equiv (\alpha,\beta) \right \}.[/tex]

The continuous dual space is the following set of conjugate linear functionals, each identified with an element of [itex]\cal{H}[/itex] via the conjugate linear analogue of an isomorphism (anti-isomorphism?), defined by the inner product:

[tex]\cal{H}'=\left \{ F_\alpha \text{ continuous } \; | \; F_\alpha(\beta) \equiv (\beta,\alpha) \right \}.[/tex]

Is Ballentine's terminology exceptional? Other sources use the name "(continuous) dual space" together with the symbol [itex]\cal{H}^{\times}[/itex] or [itex]\cal{H}^*[/itex] for what Ballentine calls "the conjugate space".

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# Continuous dual space and conjugate space

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