Talk:Injective metric space

message: Thanks David, for making that correction.  That was fast! Vegasprof 22:24, 8 November 2006 (UTC)
thread_title:  Thanks 

message: David, here is a partial answer to your question (it still might be more than you care for), a partial list of my old results. Truly metric spaces are meant, with fixed metric function each (as opposed to metrisable spaces). I'll omit my results on strongly convex metric spaces, since they are less characteristic for the topic at hand:
* 1961 (early in the year) – There does not exist a σ-compact space, which contains an isometric image of every compact, 0-dimensional, metric space of diameter < 1. That was a solution to a long standing problem. I've sent my manuscript to the Marcinkiewicz's competition, in 1961 (which I won), so that it has an official status of an obscure publication :-)
* 1961 – There exists a σ-compact space, which contains an isometric image of every countable metric space. (I proved a bit more, if you were interested; these results are mostly unpublished).
* 1967 (published) – Examples of m-hyperconvex spaces which are not (m+1)-hyperconvex. (I believe that that was an answer to an Aronszajn and Panitchpakdi question; it's basic anyway).
* 1968 (published) – Theorem  Banach injective envelope of a Banach space is essentially its metric injective envelope. This allowed me to reduce Figel's theorem about linearisation of isometric embeddings of Banach spaces to my earlier, 1966 theorem about the special case of spaces C(X) (that theorem had some extras about the effect on X and Y).
Remark   For every Banach space there exists an isometric embedding into another one, such that
no three points of the image belong to any common (affine) straight line. But if you compose an isometric emebdding of Banach spaces with a proper metric, linear homomorphism (operator) into another banach space then you will get an affine (continuous) map (or simpply linear, if image of 0 umnder the isometric embedding was 0).
* 1969 (published; actually 1963-1965)   Paragraph 3 of my "Lattices with real numbers as additive operators" is purely metric. I adopt Mazur-Ulam constructions from Banach spaces for general metric spaces in order to define the central middle  s(A)  for every bounded subset A  of a metric space (it doesn't have to exist). This leads to absolutely central spaces, in which the central middle exists for every non-empty bounded subset; to strongly central spaces, when the central middle exists for every totally bounded subset; and to central spaces, when the central middle exists for every pair of points. It's easy to show that every injective metric space is absolutely central (Theorem 3.1). However I show that the operation  s(A)  is uniformly continuous w.r. to  A  (with respect to Hausdorff metric), when restricted to subsets of diameter bounded by an arbitrary constant α. Furthermore, for sets of diameter bounded by a constant I provide a better central middle function  αs(A),   which turns out to be a metric map. I apply this metric research to study d-lattices (in particular Kaplansky's translational lattices).
* 1972   Isometric embeddings of finite metric spaces into the (injective and arbitrary) finite dimensional Banach spaces. I announced my (quite complete!) results years later in AMS Notices.
* PAMS from 70-ties or 80-ties   I give my old (ooooold) result, in which every compact metric space is shown to be a metric image of the Cantor set, represented as the cartesian product of finite metric spaces (with constant non-zero distances), and belonging to the same class in terms of local capacity as the given compact space. In this sense the properly represented Cantor sets are initial spaces for all compact spaces within their classes.
:This implies the classical topological theorem (about every metric compact set being a continuous image of the Cantor set) in a conceptual, metric structural way, with a potential for applications to approximation theory. It also gives a simple, instant proof of the metric universality of the Banach function space C(Cantor), hence of C(I) too. (I still have to find the exact coordinates of my PAMS paper on Internet).
Regards, -- Wlod (talk) 02:39, 18 November 2007 (UTC)
thread_title:  some of my metric results