Diffeomorphisms between spheres and hyperplanes in infinite-dimensional Banach spaces

• Autorzy: Daniel Azagra
• Języki publikacji: EN

Abstrakty

EN

We prove that for every infinite-dimensional Banach space X with a Fréchet differentiable norm, the sphere $S_X$ is diffeomorphic to each closed hyperplane in X. We also prove that every infinite-dimensional Banach space Y having a (not necessarily equivalent) $C^p$ norm (with $p ∈ ℕ ∪ {∞}$)$ is $C^p$ diffeomorphic to $Y \ {0}$.

Słowa kluczowe

EN

$C^p$ smooth norm; spheres and hyperplanes in Banach spaces

Kategorie tematyczne

• 46B20: Geometry and structure of normed linear spaces
• 58B99: None of the above, but in this section

• Wydawca: Institute of Mathematics Polish Academy of Sciences
• Czasopismo: Studia Mathematica
• Rocznik: 1997
• Tom: 125
• Numer: 2
• Strony: 179-186

Daty

wydano

1997

otrzymano

1996-12-23

poprawiono

1997-03-11

Twórcy

autor

Daniel Azagra

• Departamento de Análisis Matemático, Facultad de Ciencias Matemáticas, Universidad Complutense, Madrid, 28040, Spain

Bibliografia

• [1] C. Bessaga, Every infinite-dimensional Hilbert space is diffeomorphic with its unit sphere, Bull. Acad. Polon. Sci. Sér. Sci. Math. Astronom. Phys. 14 (1966), 27-31.
• [2] C. Bessaga, Interplay between infinite-dimensional topology and functional analysis. Mappings defined by explicit formulas and their applications, Topology Proc. 19 (1994), 15-35.
• [3] C. Bessaga and A. Pełczyński, Selected Topics in Infinite-Dimensional Topology, Monograf. Mat. 58, PWN, Warszawa, 1975.
• [4] R. Deville, G. Godefroy and V. Zizler, Smoothness and Renormings in Banach Spaces, Pitman Monographs Surveys Pure and Appl. Math. 64, Longman, 1993.
• [5] T. Dobrowolski, Smooth and R-analytic negligibility of subsets and extension of homeomorphisms in Banach spaces, Studia Math. 65 (1979), 115-139.
• [6] T. Dobrowolski, Every infinite-dimensional Hilbert space is real-analytically isomorphic with its unit sphere, J. Funct. Anal. 134 (1995), 350-362.
• [7] T. Dobrowolski, Relative classification of smooth convex bodies, Bull. Acad. Polon. Sci. Sér. Sci. Math. Astronom. Phys. 25 (1977), 309-312.
• [8] B. M. Garay, Cross-sections of solution funnels in Banach spaces, Studia Math. 97 (1990), 13-26.
• [9] B. M. Garay, Deleting homeomorphisms and the failure of Peano's existence theorem in infinite-dimensional Banach spaces, Funkcial. Ekvac. 34 (1991), 85-93.
• [10] K. Goebel and J. Wośko, Making a hole in the space, Proc. Amer. Math. Soc. 114 (1992), 475-476.
• [11] W. T. Gowers, A solution to Banach's hyperplane problem, Bull. London Math. Soc. 26 (1994), 523-530.
• [12] W. T. Gowers and B. Maurey, The unconditional basic sequence problem, J. Amer. Math. Soc. 6 (1993), 851-874.
• [13] R. C. James, Weakly compact sets, Trans. Amer. Math. Soc. 113 (1964), 129-140.
• [14] K V. L. Klee, Convex bodies and periodic homeomorphisms in Hilbert space, ibid. 74 (1953), 10-43.
• [15] S. Troyanski, On locally uniformly convex and differentiable norms in certain non-separable Banach spaces, Studia Math. 37 (1971), 173-180.