Thus, for instance Kashin's theorem becomes the statement that every nonstandard finite dimensional Banach space of some nonstandard finite dimension N has a half-dimensional an M-dimensional subspace which is isomorphic (in the category of bounded nonstandard linear transformations) to $\ell^2$.\ell^2(M)$whenever$M \leq N/2$(or more generally when$M \leq (1-\epsilon)N$for some standard$\epsilon > 0$. Dvoretsky's theorem is trickier. Here, I guess one needs to work with the category of almost isometriescontractions: operators whose operator norm, and norm of inverse, is at most$1+o(1)$(i.e. between$1-\epsilon$and bounded by$1+\epsilon$for every standard$\epsilon > 0$. Then I think the theorem says that any nonstandard finite dimensional Banach space with some nonstandard dimension N has a M-dimensional subspace that is almost isometric to$\ell^2$with dimension at least M, \ell^2(M)$, whenever $M = o(\log N)$. (I may have messed up the quantifiers slightly, but this is pretty close to what the nonstandard translation of things.things should be.)
Thus, for instance Kashin's theorem becomes the statement that every nonstandard finite dimensional Banach space has a half-dimensional subspace which is isomorphic (in the category of bounded nonstandard linear transformations) to $\ell^2$.
Dvoretsky's theorem is trickier. Here, I guess one needs to work with the category of almost isometries: operators whose operator norm, and norm of inverse, is $1+o(1)$ (i.e. between $1-\epsilon$ and $1+\epsilon$ for every standard $\epsilon > 0$. Then I think the theorem says that any nonstandard finite dimensional Banach space with some nonstandard dimension N has a subspace that is almost isometric to $\ell^2$ with dimension at least M, whenever $M = o(\log N)$. (I may have messed up the quantifiers slightly, but this is pretty close to the nonstandard translation of things.)