Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. Join them; it only takes a minute:

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

The discussion in Convergence in topologies, especially the comments of GEdgar, led me to another (converse) question concerning convergence. In the paper

G. A. Edgar, A long James space, in: Measure Theory, Oberwolfach 1979, Lectures Notes in Math. 794, Springer-Verlag (1980) p. 31-37

he points out that the weak and weak* topologies in $J(\omega_1)^*$ have different Borel $\sigma$-fields. Do they have the same convergent sequences? Heuristically, they should as $J(\omega_1)$ just "differs" with a reflexive space "by one dimension".

share|cite|improve this question

Impossible, since this space contains a copy of James space which is separable but not reflexive (every separable Grothendieck space, that is, space with this property must be reflexive).

share|cite|improve this answer

Tomek's answer enlarged: Even in $J = J(\omega)$ there is a sequence that converges weak* but not weakly. Explictly: in the notation of my paper cited, sequence $e_n$ converges weak* to $e_\omega$, but not weakly. So if we alternate $e_1, e_\omega, e_2, e_\omega, e_3, e_\omega, \dots$ we get a sequence that converges weak* but not weakly.

share|cite|improve this answer

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.