Une remarque sur l'homéomorphie des champs fonctionnels

Mazur, S.

doi:10.4064/sm-1-1-83-85

Cited by 75 publications

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“…But the latter two spaces are known to be homeomorphic with Hubert space (by results of Mazur [12] and Kadee [ó]) and the desired conclusion follows. Now suppose B is a separable conjugate space or, more generally, that B is a norm-separable w*-closed linear subspace of a conjugate Banach space L*.…”

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confidence: 81%

Topological equivalence of a Banach space with its unit cell

Klee¹

1961

Bull. Amer. Math. Soc.

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Several years ago [8] we proved that Hubert space is homeomorphic with both its unit sphere {#: ||x|| = 1} and its unit cell {x:||x||^l}. Later [9] we showed that in every infinite-dimensional normed linear space, the unit sphere is homeomorphic with a (closed) hyperplane and the unit cell with a closed halfspace. It seems probable that every infinite-dimensional normed linear space is homeomorphic with both its unit sphere and its unit cell, but the question is unsettled even for Banach spaces. Corson [4] has recently proved that every fc$ 0 -dimensional normed linear space is homeomorphic with its unit cell. In the present note, we establish the same result for a class of infinite-dimensional Banach spaces which is believed to include all such spaces. It is proved to include every infinite-dimensional Banach space which is reflexive, or admits an unconditional basis, or is a separable conjugate space, or is a space CM of all bounded continuous real-valued functions on a metric space M.We employ the following tools:(1) If E and F are Banach spaces and u is a continuous linear transformation of E onto F, then there exist a constant ra£]0, <» [ and continuous mapping v of F into E such that uvx = x, vrx = rvx, and \\vx\\ rSw||#|| for all x£F and rÇiR (the real number space). If G is the kernel of u and hy= (uy, vuy-y)(E:FXG for each yÇzE, then h is a homeomorphism of E onto FXG. Let \$p, q)\\ =max (\\p\\, \\q\$ for all (p, q)GFXG, and let £y = (|M|/||ft:y||)A:y for all y£E. Then £ is a homeomorphism of E onto FXG which carries the unit cell of E onto that of FXG.(2) If S is a closed linear subspace of a Banach space £, then E is homeomorphic with the product space (E/S) XS and the unit cell of E is homeomorphic with the unit cell of this product space (with respect to any norm compatible with the product topology).(3) In each infinite-dimensional normed linear space, the unit cell is homeomorphic with a closed halfspace.(4) If Q is an open halfspace in an infinite-dimensional normed linear space and p is a point in the boundary of Q, then QVJ {p} is homeomorphic with Q.

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confidence: 81%

Topological equivalence of a Banach space with its unit cell

Klee¹

1961

Bull. Amer. Math. Soc.

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“…This is achieved by the so-called Mazur map first used by Mazur in 1929 [99]. We define ϕ : It was already known that the cotype assumption here is necessary by a result of Raynaud [115]: see Theorem 4.12 below.…”

Section: Uniform and Coarse Embeddings In L P -Spacesmentioning

confidence: 99%

The Nonlinear Geometry of Banach Spaces

Kalton¹

2008

Rev. Mat. Complut.

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“…Let 2 ≤ p < q < ∞. Lindenstrauss [3] proved that L p is not uniformly homeomorphic to L q , while the Mazur map [4] shows that B Lp is uniformly homeomorphic to B Lq , and S LP is uniformly homeomorphic to S Lq . Therefore, the fact that the unit balls or the spheres of two normed spaces are uniformly homeomorphic does not imply that the spaces are also uniformly homeomorphic.…”

Section: Introductionmentioning

confidence: 99%