Cardinality of the metric projection on typical compact sets in Hilbert spaces

In the first and second parts of this sequence we dealt with applications of graph theory to distance distribution in certain sets in euclidean spaces, to potential theory, to estimations of the transfinite diameter [1] and to value distribution of "triangle functional " (e.g. perimeter, area of triangles) [2]. The basic tool is provided in all these applications by the result formulated as Lemma 2. This, an essentially pure logical result, proves to be a very flexible and versatile instrument in applications.Here the same method is used in an abstract setting. First we deduce certain results for the density of a given family of subsets of an abstract set S in another family of subsets of the same S. Then we apply the results obtained to distance distribution in certain (e.g. totally bounded or compact) sets in metric spaces, in particular in a normed linear function space. Applications of this method to functional on Hilbert spaces were given by Katona [3].

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Gaussian Hilbert Spaces

10.1017/cbo9780511526169 ◽

1997 ◽

Cited By ~ 257

Author(s):

Svante Janson

Keyword(s):

Hilbert Spaces

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A closer look at the stable completion

10.23943/princeton/9780691161686.003.0005 ◽

2017 ◽

Author(s):

Ehud Hrushovski ◽

François Loeser

Keyword(s):

Inverse Limit ◽

Unit Vector ◽

Vector Spaces ◽

Dimensional Vector ◽

Compact Sets ◽

Linear Topology ◽

Topological Embedding ◽

Definable Set ◽

Finite Dimensional ◽

The One

This chapter introduces the concept of stable completion and provides a concrete representation of unit vector Mathematical Double-Struck Capital A superscript n in terms of spaces of semi-lattices, with particular emphasis on the frontier between the definable and the topological categories. It begins by constructing a topological embedding of unit vector Mathematical Double-Struck Capital A superscript n into the inverse limit of a system of spaces of semi-lattices L(Hsubscript d) endowed with the linear topology, where Hsubscript d are finite-dimensional vector spaces. The description is extended to the projective setting. The linear topology is then related to the one induced by the finite level morphism L(Hsubscript d). The chapter also considers the condition that if a definable set in L(Hsubscript d) is an intersection of relatively compact sets, then it is itself relatively compact.

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