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Added third paragraph.
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Bill Johnson
Bill Johnson
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Kevin, there are non reflexive spaces with non trivial type--even of type 2. James constructed the first one; his argument is very complicated. Later Pisier-Xu did it much more simply using interpolation between $\ell_1$ and $\ell_\infty$, but using the universal non weakly compact operator instead of the formal identity between the two spaces. See

Random series in the real interpolation spaces between the spaces vp. Geometrical aspects of functional analysis (1985/86), 185–209, Lecture Notes in Math., 1267, Springer, Berlin, 1987.

For a reflexive space with non-trivial type that is not superreflexive take the $\ell_2$ sum of all finite dimensional subspaces of the Pisier-Xu space.

Kevin, there are reflexive spaces with non trivial type--even of type 2. James constructed the first one; his argument is very complicated. Later Pisier-Xu did it much more simply using interpolation between $\ell_1$ and $\ell_\infty$, but using the universal non weakly compact operator instead of the formal identity between the two spaces. See

Random series in the real interpolation spaces between the spaces vp. Geometrical aspects of functional analysis (1985/86), 185–209, Lecture Notes in Math., 1267, Springer, Berlin, 1987.

Kevin, there are non reflexive spaces with non trivial type--even of type 2. James constructed the first one; his argument is very complicated. Later Pisier-Xu did it much more simply using interpolation between $\ell_1$ and $\ell_\infty$, but using the universal non weakly compact operator instead of the formal identity between the two spaces. See

Random series in the real interpolation spaces between the spaces vp. Geometrical aspects of functional analysis (1985/86), 185–209, Lecture Notes in Math., 1267, Springer, Berlin, 1987.

For a reflexive space with non-trivial type that is not superreflexive take the $\ell_2$ sum of all finite dimensional subspaces of the Pisier-Xu space.

Source Link
Bill Johnson
Bill Johnson
  • 31.5k
  • 5
  • 89
  • 138

Kevin, there are reflexive spaces with non trivial type--even of type 2. James constructed the first one; his argument is very complicated. Later Pisier-Xu did it much more simply using interpolation between $\ell_1$ and $\ell_\infty$, but using the universal non weakly compact operator instead of the formal identity between the two spaces. See

Random series in the real interpolation spaces between the spaces vp. Geometrical aspects of functional analysis (1985/86), 185–209, Lecture Notes in Math., 1267, Springer, Berlin, 1987.