Analytic and $C^k$ approximations of norms in separable Banach spaces

• Autorzy: Robert Deville , Vladimir Fonf , Petr Hájek
• Języki publikacji: EN

Abstrakty

EN

We prove that in separable Hilbert spaces, in $ℓ_{p}(ℕ)$ for p an even integer, and in $L_{p}[0,1]$ for p an even integer, every equivalent norm can be approximated uniformly on bounded sets by analytic norms. In $ℓ_{p}(ℕ)$ and in $L_{p}[0,1]$ for p ∉ ℕ (resp. for p an odd integer), every equivalent norm can be approximated uniformly on bounded sets by $C^[p]}$-smooth norms (resp. by $C^{p-1}$-smooth norms).

Słowa kluczowe

EN

analytic norm; approximation; convex function; geometry of Banach spaces

Kategorie tematyczne

• 46G20: Infinite-dimensional holomorphy
• 46B20: Geometry and structure of normed linear spaces

• Wydawca: Institute of Mathematics Polish Academy of Sciences
• Czasopismo: Studia Mathematica
• Rocznik: 1996
• Tom: 120
• Numer: 1
• Strony: 61-74

Daty

wydano

1996

otrzymano

1995-09-11

poprawiono

1996-01-19

Twórcy

autor

Robert Deville

• Department of Mathematics, Université de Bordeaux, 351, Cours de la Libération, 33400 Talence, France,

autor

Vladimir Fonf

• Department of Mathematics and Computer Sciences, Ben Gurion University of the Negev, Beer Sheva, Israel,

autor

Petr Hájek

• Department of Mathematics, University of Alberta, Edmonton, Alberta T6G 2G1, Canada,

Bibliografia

• [D] R. Deville, Geometrical implications of the existence of very smooth bump functions in Banach spaces, Israel J. Math. 6 (1989), 1-22.
• [DFH] R. Deville, V. Fonf and P. Hájek, Analytic and polyhedral approximation of convex bodies in separable polyhedral Banach spaces, to appear.
• [DGZ] R. Deville, G. Godefroy and V. Zizler, Smoothness and Renormings in Banach Spaces, Pitman Monographs Surveys Pure Appl. Math. 64, Longman, 1993.
• [Dieu] J. Dieudonné, Foundations of Modern Analysis, Academic Press, New York, 1960.
• [FFWZ] M. Fabian, D. Preiss, J. H. M. Whitfield and V. Zizler, Separating polynomials on Banach spaces, Quart. J. Math. Oxford Ser. (2) 40 (1989), 409-422.
• [Fe] H. Federer, Geometric Measure Theory, Springer, 1969.
• [HH] P. Habala and P. Hájek, Stabilization of polynomials, C. R. Acad. Sci. Paris Sér. I 320 (1995), 821-825.
• [H1] R. Haydon, Normes infiniment différentiables sur certains espaces de Banach, ibid. 315 (1992), 1175-1178.
• [H2] R. Haydon, Smooth functions and partitions of unity on certain Banach spaces, to appear.
• [Ku1] J. Kurzweil, On approximation in real Banach spaces, Studia Math. 14 (1954), 213-231.
• [Ku2] J. Kurzweil, On approximation in real Banach spaces by analytic operations, ibid. 16 (1957), 124-129.
• [M] P. Mazet, Analytic Sets in Locally Convex Spaces, North-Holland Math. Stud. 89, North-Holland, 1984.
• [N] L. Nachbin, Topology on Spaces of Holomorphic Mappings, Springer, 1969.
• [NS] A. M. Nemirovskiĭ and S. M. Semenov, On polynomial approximation of functions on Hilbert space, Mat. Sb. 92 (1973), 257-281 (in Russian).
• [Zp] M. Zippin, The separable extension problem, Israel J. Math. 26 (1977), 372-387.