Geometric Theory of Spaces of Integral Polynomials and Symmetric Tensor Products
By C. Boyd, R. Ryan. 2001
51 citations recorded by the source corpus.
Abstract We investigate the nature of extreme, (weak*-) exposed, and (weak*-) strongly exposed points of the unit ball of spaces of n-homogeneous integral polynomials on, and n-fold symmetric products of, a Banach space E. For the space of integral polynomials we show the set of extreme points is contained in the set {±φn: φ∈E′, ‖φ‖=1}. We give Smul'yan type theorems for spaces of n-homogeneous polynomials and n-fold symmetric tensors that characterise weak*-exposed (resp. weak*-strongly exposed) points in terms of Gâteaux (resp. Frechet) differentiability of the norm on various spaces of tensor products and polynomials. Our study of the geometry of these spaces has many applications: When E has the Radon–Nikodým property we show that the spaces of n-homogeneous integral and nuclear polynomials are isomerically isomorphic for each integer n. When the dimensions of E and n are both at least 2 then the space of n-homogeneous polynomials on E is neither smooth nor rotund. For a certain class of reflexive Banach space the space of n-homogeneous approximable polynomials on E is either reflexive or is not isometric to a dual Banach space. We conclude with a Choquet Theorem for a space of homogeneous polynomials.
Subfields: Applied Mathematics, Geometry and Topology, Mathematical Physics
View Paper: https://www.semanticscholar.org/paper/c297f8ea7f7855a657a4d5d8d3fe6c950888befb
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In Applied Mathematics
In Geometry and Topology
In Mathematical Physics
Citations in the corpus
- Comparing different classes of absolutely summing multilinear operators (2005, background): “This is the case of weakly compact, completely continuous and unconditional converging operators ([3], [5], [24], [25], [29], [34], [35] or [49]), the case of integral and nuclear operators ([1], [ 8 ], [12], [18] or [50]) and the case of Hilbert-Schmidt operators ([13], [20] or [31]).”