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I am reading the paper

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

and I am pretty confused by the remarks after the proof of Proposition 3.

Is it clear that $J(\omega_1)$ is of codimension 1 in $J(\omega_1)^{** }$ (via the canonical map) in the same way as the (usual) James space $J$ is of codimension 1 in $J^{** }$?

Btw. I posted this question at MathStack but it has not been answered.

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  • $\begingroup$ By now, math.stackexchange.com gets questions at such a rate that--even if I look there every day--I may miss questions of interest to me. $\endgroup$
    – Gerald Edgar
    Commented Oct 20, 2011 at 1:45

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Bill is correct: $J(\omega_1)$ is not of codimension $1$ in its bidual. The remarks after Proposition 3 say: (if $\eta$ is infinite) then $J(\eta)^{**}$ is isometric to $\widetilde{J}(\eta+1)$, and the set-theoretic inclusion is the canonical embedding. The tilde on the $J$ means that we drop the requirement of continuity. So we get a new dimension for each limit ordinal up to $\eta$. Basically, while elements of $J(\eta)$ must be continuous, elements of the bidual $\widetilde{J}(\eta+1)$ need not be continuous, so we can have "jump" discontinuities at the limit ordinals.

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  • $\begingroup$ This other question on the long James space might be of interest to you, too: math.stackexchange.com/questions/69807 $\endgroup$
    – Theo Buehler
    Commented Oct 20, 2011 at 5:31
  • $\begingroup$ Btw. is $J(\eta)$ isomorphic to $J(\xi)$ whenever $\omega\leq \xi, \eta<\omega_1$? I suspect not. $\endgroup$
    – Briannon
    Commented Oct 20, 2011 at 19:31
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I don't have access here to Edgar's article, but I very much doubt that he said that the long James space is quasi-reflexive!

You can get an explicit description and discussion of the second dual of the long James space in two papers from the mid 1980s:

Zhao, Jun Feng(PRC-WUHAN) The transfinite basis of the bidual space of the long James space. Acta Math. Sci. (English Ed.) 5 (1985), no. 3, 295–301. 46B10

MR0843290 (87k:46044) Zhao, Jun Feng The ordering structure on Banach spaces. II. (Chinese. English summary) J. Wuhan Univ. Natur. Sci. Ed. 1985, no. 3, 11–18. 46B20

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