On Topological Classification of Non-Separable Banach Spaces

Bessaga, C.; Kadec, M. I.

doi:10.1515/9781400881406-003

Cited by 39 publications

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“…The possibility of multiplying elements of a group by reals in a homomorphic way and the convex structure of a group facilitate investigations of such groups and has many deep topological and geometric consequences. For example, every normed vector space is an absolute extensor for metric spaces ([5]; see also [2, Theorem II.3.1]), locally compact normed vector spaces are finite dimensional and are uniquely determined (up to topological group isomorphism) by their dimension, non-locally compact completely metrizable normed spaces (that is, infinitedimensional Banach spaces [9]) as topological spaces are also uniquely determined P. Niemiec (B) …”

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confidence: 99%

Normed Topological Pseudovector Groups

Niemiec

2010

Appl Categor Struct

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A normed topological pseudovector group (NTPVG for short) is a valued topological group (V, +, · ) (not necessarily Abelian) endowed with a continuous scalar multiplicationt · x = t x for each t, s ∈ R + and x, y ∈ V. It is shown that every valued topological group can be isometrically and group-homomorphically embedded in a NTPVG as a closed subset by means of a functor. Locally compact NTPV groups are fully classified. It is shown that the (unbounded) Urysohn universal metric space can be endowed with a structure of a NTPV group of exponent 2.Keywords Normed vector spaces · Topological groups · Locally compact groups · Valued groups · Dynamical systems · Urysohn universal metric space Mathematics Subject Classifications (2010) 22A99 · 46B99 · 46A99Normed vector spaces are metrizable topological groups of a very special kind. One may say that these are groups 'modelled' on the additive group of real numbers. The possibility of multiplying elements of a group by reals in a homomorphic way and the convex structure of a group facilitate investigations of such groups and has many deep topological and geometric consequences. For example, every normed vector space is an absolute extensor for metric spaces ([5]; see also [2, Theorem II.3.1]), locally compact normed vector spaces are finite dimensional and are uniquely determined (up to topological group isomorphism) by their dimension, non-locally compact completely metrizable normed spaces (that is, infinitedimensional Banach spaces [9]) as topological spaces are also uniquely determined P. Niemiec (B)

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confidence: 99%

Normed Topological Pseudovector Groups

Niemiec

2010

Appl Categor Struct

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“…Also in large classes of nonseparable Banach spaces (perhaps all) the density character of a Banach space is its only topological invariant (see the book [2] for details). The situation changes considerably if we consider uniform homeomorphisms.…”

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confidence: 99%

Uniform equivalence between Banach spaces

Aharoni¹,

Lindenstrauss²

1978

Bull. Amer. Math. Soc.

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It is a well-known result of Kadec that every two separable infinite dimensional Banach spaces are homeomorphic. Also in large classes of nonseparable Banach spaces (perhaps all) the density character of a Banach space is its only topological invariant (see the book [2] for details). The situation changes considerably if we consider uniform homeomorphisms. Several results are known which prove the nonexistence of uniform homeomorphisms between certain Banach spaces of the same density character. As a matter of fact, the following problem was raised by many mathematicians: Do there exist two nonisomorphic Banach spaces which are uniformly homeomorphic? (Le. does the uniform structure of a Banach space determine its linear structure?) For a recent survey of results related to this problem see [3].While studying the question of existence of nonlinear liftings, we found in a surprisingly simple manner an example which answers this problem. Let T be a set of the cardinality of the continuum. Then c 0 (T) is lipschitz equivalent to a certain closed subspace X of l^ (i.e. there is a map T from c 0 (r) onto X so that T and T~x satisfy a lipschitz condition). Since there is no sequence of continuous linear functionals which separate the points in c 0 (T), this space is not isomorphic to a subspace of /«.Let UD F be Banach spaces and let admits a lipschitz (resp. uniformly continuous) lifting if there is a lipschitz (resp. uniformly continuous) map \p: UjV-• C/so that

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“…The invertibility of infinite-dimensional complete normed spaces should not be surprising. Unlike the finite dimensional case, every infinite-dimensional Banach space E is homeomorphic to its unit sphere S [14,3]. A key ingredient of the proof is the topological equivalence L L × R for every infinite-dimensional Banach space L. The assertion will follow from this since S is homeomorphic to an (infinite-dimensional) closed hyperplane L of E which is in turn homeomorphic to L × R E (see [3, p. 190]).…”

Section: Corollary 8 Every Infinite-dimensional Normed Space N Is Inmentioning

confidence: 99%

Invertibility in infinite-dimensional spaces

Tseng

Wong

1999

Proc. Amer. Math. Soc.

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Abstract. An interesting result of Doyle and Hocking states that a topological n-manifold is invertible if and only if it is a homeomorphic image of the n-sphere S n . We shall prove that the sphere of any infinite-dimensional normed space is invertible. We shall also discuss the invertibility of other infinite-dimensional objects as well as an infinite-dimensional version of the Doyle-Hocking theorem.

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On Topological Classification of Non-Separable Banach Spaces

Cited by 39 publications

References 0 publications

Normed Topological Pseudovector Groups

Normed Topological Pseudovector Groups

Uniform equivalence between Banach spaces

Invertibility in infinite-dimensional spaces

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