By Antonio J. Guirao, Vicente Montesinos, Václav Zizler
This is an number of a few easily-formulated difficulties that stay open within the examine of the geometry and research of Banach areas. Assuming the reader has a operating familiarity with the fundamental result of Banach area thought, the authors specialize in suggestions of simple linear geometry, convexity, approximation, optimization, differentiability, renormings, vulnerable compact producing, Schauder bases and biorthogonal platforms, fastened issues, topology and nonlinear geometry.
The major goal of this paintings is to assist in convincing younger researchers in sensible research that the idea of Banach areas is a fertile box of study, choked with attention-grabbing open difficulties. contained in the Banach house zone, the textual content can assist disclose younger researchers to the intensity and breadth of the paintings that is still, and to supply the viewpoint essential to decide on a course for extra study.
Some of the issues are longstanding open difficulties, a few are contemporary, a few are extra very important and a few are just neighborhood difficulties. a few will require new principles, a few might be resolved with just a refined mix of identified evidence. despite their foundation or toughness, each one of those difficulties records the necessity for additional learn during this area.
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Extra info for Open Problems in the Geometry and Analysis of Banach Spaces
Example text
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. V. , to [FLP01]). Note that the centers of these shifts in Klee’s result clearly form a Chebyshev set that is not convex. It is simple to prove that every closed convex subset of a strictly convex reflexive Banach space is Chebyshev.
The theory of biorthogonal systems is crucial for understanding the structure of Banach spaces, in particular of nonseparable ones. Many problems in this area are widely open. , they use special axioms of Set Theory. , [HMVZ08, pp. 148 and 152] or [To06]. 1 be a collection of open and dense sets in K. 1 U˛ is dense in K. By considering the complements of singletons in Œ0; 1, we see that this axiom, in the setting of Baire category, contradicts the Continuum Hypothesis. 1 and yet consistent with ZFC.
Note that if X has a shrinking FDD then X is separable. The following seems to be an open problem: Problem 110. Assume that X is a separable Banach space such that X is nonseparable. Does there exist a subspace Y of X such that both Y and X=Y have nonseparable dual and both have finite-dimensional decompositions? We refer again to [LinTza77, Sect. g] and also to [DGHZ88], where the result is proved for spaces that contain isomorphic copies of `1 . We note in passing that J. Hagler in [Ha87] showed that if X is a separable Banach space with nonseparable dual, then X contains a subspace with Schauder basis and nonseparable dual.