On the Dimension of Almost Hilbertian Subspaces of Quotient Spaces
نویسنده
چکیده
The question of the dimension of almost Hilbertian subspaces is resolved in [1] where it is shown that every Banach space E of dimension n possesses almost Hilbertian subspaces of dimension c(logn), where c is an absolute constant, and that this estimate is the best possible. When the net is spread wider to include quotient spaces and subspaces of quotient spaces we should expect to find instances of almost Hilbertian spaces of much higher dimension; it comes as no surprise to infer at once, from [1, Theorem 2.9], that E possesses either almost Hilbertian subspaces or almost Hilbertian quotient spaces of dimension at least en. The main theorem of this article (Theorem 5) is a result of this type. The question broached here also receives consideration in [2], where similar results are obtained by different methods. The proofs will follow readily from some results on the volume of convex bodies in IR", of which the oldest is an inequality of Santalo [4]. Volumetric considerations have recently held special significance for Banach-space theorists. In [7] the authors proved the existence of so-called Kashin decompositions for some families of Banach spaces, and they were led to define a certain affine invariant associated with an arbitrary finite-dimensional space which they called the volume ratio. We shall use a theorem relating this quantity to the existence of almost Hilbertian subspaces.
منابع مشابه
Euclidean spaces as weak tangents of infinitesimally Hilbertian metric spaces with Ricci curvature bounded below
We show that in any infinitesimally Hilbertian CD∗(K,N)-space at almost every point there exists a Euclidean weak tangent, i.e. there exists a sequence of dilations of the space that converges to a Euclidean space in the pointed measured Gromov-Hausdorff topology. The proof follows by considering iterated tangents and the splitting theorem for infinitesimally Hilbertian CD∗(0, N)-spaces.
متن کاملSymplectic geometry on symplectic knot spaces
Symplectic knot spaces are the spaces of symplectic subspaces in a symplectic manifold M . We introduce a symplectic structure and show that the structure can be also obtained by the symplectic quotient method. We explain the correspondence between coisotropic submanifolds in M and Lagrangians in the symplectic knot space. We also define an almost complex structure on the symplectic knot space,...
متن کاملSubspace structure of some operator and Banach spaces
We construct a family of separable Hilbertian operator spaces, such that the relation of complete isomorphism between the subspaces of each member of this family is complete Kσ . We also investigate some interesting properties of completely unconditional bases of the spaces from this family. In the Banach space setting, we construct a space for which the relation of isometry of subspaces is equ...
متن کاملAn Example of an Asymptotically Hilbertian Space Which Fails the Approximation Property
Following Davie’s example of a Banach space failing the approximation property ([D]), we show how to construct a Banach space E which is asymptotically Hilbertian and fails the approximation property. Moreover, the space E is shown to be a subspace of a space with an unconditional basis which is “almost” a weak Hilbert space and which can be written as the direct sum of two subspaces all of who...
متن کامل