James' theorem states that a Banach space $B$ is reflexive iff every bounded linear functional on $B$ attains its maximum on the closed unit ball in $B$.

Now I wonder if I can drop the constraint that it is a ball and replace it by "convex set". That is, I want to know if every bounded linear functional on a reflexive Banach space $B$ attains its maximum on a closed and bounded convex set in $B$.

By Pietro's answer this is known to be true. Is the maximum unique? In optimization by vector space methods it is know that this is true if the set is the closed ball. This was actually my biggest question since I want to show a optimization problem has a unique solution.

  • $\begingroup$ +1 not particularly for the question but because that theorem is something I'd been wondering about recently but didn't know that someone had proven it! Don't happen to have a reference for it, do you? $\endgroup$ – Loop Space Sep 2 '10 at 11:05
  • $\begingroup$ This seems to be James' article about this: springerlink.com/content/07411341j5792482 $\endgroup$ – Jonas Teuwen Sep 2 '10 at 11:11
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    $\begingroup$ Uniqueness seems to fail even in finite dimensional cases (think of $R^2$ with sup norm) and might be related to uniform convexity -- just a guess $\endgroup$ – Piero D'Ancona Sep 2 '10 at 12:16
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    $\begingroup$ Uniqueness is equivalent to strict convexity of the ball, Piero: If $f$ attains its max at two points $x$ and $y$, then it attains it at everything on the line segment joining $x$ and $y$. On the other hand, if the sphere contains a line segment, you can define a norm one linear functional $f$ on the two dimensional subspace spanned by the endpoints of the segment s.t. the segment is contained in $[f=1]$ and use Hahn-Banach to extend the functional to the entire space. $\endgroup$ – Bill Johnson Sep 2 '10 at 14:06
  • $\begingroup$ Indeed for every Banach space $X$ and every bounded linear functional $f$, there is an equivalent norm of $X$ producing a unit ball such that $f$ has multiple maximum points there (just take $\max\{\|x\|,\epsilon |\langle f,x\rangle |\}$) $\endgroup$ – Pietro Majer Sep 2 '10 at 14:33

A closed and bounded convex set of a reflexive Banach is w* compact, hence any bounded linear functional does attain its maximum and minimum there. On the other direction, the presence of a closed bounded convex set on which all bounded linear functionals have their maximum, of course, says nothing on the reflexivity of the space (the convex could be a single point).

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    $\begingroup$ Ah, right, that a bounded closed convex set in a reflexive Banach space is wk*-compact is a corollary of Alaoglu's theorem and Hahn-Banach. Thanks! $\endgroup$ – Jonas Teuwen Sep 2 '10 at 10:50
  • $\begingroup$ By the way, I was also wondering if this maximum is unique (I edited the post). $\endgroup$ – Jonas Teuwen Sep 2 '10 at 11:18
  • $\begingroup$ And what about the closure of open bounded convex nonempty sets? $\endgroup$ – Piero D'Ancona Sep 2 '10 at 12:01
  • $\begingroup$ eh, that's what I was also wondering... I think one should be able to reduce to the case of a true ball -after all that thing is almost a ball: it just lacks the symmetry B=-B. $\endgroup$ – Pietro Majer Sep 2 '10 at 14:36

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