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I am reviewing the proof of James theorem, i.e. a Banach space is reflexive iff every continuous linear functional obtains its norm. Every thing I find online shows one direction ($\Leftarrow$), but not ($\Rightarrow$). I am having issues seeing it and any help would be appreciated.

Scott
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2 Answers2

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If $f$ is a continuous linear functional on a reflexive space $X$ the it is continuous when $X$ is given the weak topology. The closed unit ball of $X$ is weakly compact (by Banach Alaoglu Theorem) so $f$ attains its maximum on the ball. This maximum is, of course, the norm of $f$.

Kavi Rama Murthy
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I don't have enough reputation to add a comment on Kavi's answer, but the Banach-Alaoglu Theorem claims that the unit ball of the dual of a Banach space is compact in the weak* topology, rather than that the unit ball of a Banach space is comapct in the weak topology, which is in general false. In fact, a Banach space is reflexive if and only if its unit ball is weakly compact. If a Banach space $X$ is reflexive, then, since a continuous linear functional $f\in X^*$ is continuous $(X, \text{weak})\to \mathbb R$, the set $f(B_X)$ is compact, hence has and contains a supremum.

David Farrell
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  • Kavi Rama Murthy assumed that $X$ is reflexive, thus there's no problem with the unit ball being weakly compact. – Jakobian Jun 14 '23 at 21:39