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.



Codimension of $J(\omega_1)$ in its bidual
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.


