By Milgram R. (ed.)

ISBN-10: 0821814338

ISBN-13: 9780821814338

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**Extra resources for Algebraic and Geometric Topology, Part 2**

**Sample text**

In the following problem, X will be a real Banach space. We need a few definitions. First, if X is a real Banach space and T W X ! X/ denotes the set of all bounded operators from X into X. X/ D 0 then. However, the following seems to be open: Problem 78 (D. Li, C. Finet). `p / > 0 for all p 6D 2? We followed here [KMP00]. 1 Chebyshev Sets A subset K of a Banach space X is said to be a Chebyshev set if every point in X has a unique nearest point in K. In such a case, the mapping that to x 2 X associates the point in K at minimum distance is called the metric projection.

I. Aharoni and J. , [FHHMZ11, p. I/ for any uncountable I. I/, we get an example of two nonseparable spaces that are Lipschitz homeomorphic and not linearly isomorphic. We will come to these questions later on again. 40 2 Basic Linear Geometry Due to the power of the concept of differentiability, the situation in separable spaces is completely different. This is due to the following result of G. Godefroy and N. J. x2 /k=kx1 x2 k W x1 ; x2 2 X; x1 6D x2 g. , by a projection of norm 1. X/ is Y. 0/ D 0.

543]). J. Lindenstrauss used it in a substantial strengthening of the Krein–Milman theorem [Lin63]. J. Lindenstrauss [Lin63] and independently E. Asplund [As68] used it to show the Fréchet differentiability at dense sets of points of continuous convex functions on separable reflexive spaces. S. L. , [FHHMZ11, p. 587]). S. L. Troyanski and, independently, J. , [DeGoZi93b], [FHHMZ11, p. 409], and [HMVZ08]). M. I. Kadets and S. L. 8 Rotund Renormings of Banach Spaces 47 conversely, every space that admits no LUR norm must contain a copy of `1 .