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Filling analytic sets by the derivatives of  -smooth bumps
Author(s):
Marián
Fabian;
Ondrej
F. K.
Kalenda;
Jan
Kolár
Journal:
Proc. Amer. Math. Soc.
133
(2005),
295-303.
MSC (2000):
Primary 54H05;
Secondary 58C25, 46G05
Posted:
August 24, 2004
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Abstract:
If  is an infinite-dimensional Banach space, with separable dual, and  is an analytic set such that any point  can be reached from  by a continuous path contained (except for the point  ) in the interior of  , then  is the range of the derivative of a  -smooth function on  with bounded nonempty support.
References:
-
- 1.
- D. Azagra and R. Deville, James' theorem fails for starlike bodies, Journal of Functional Analysis 180, 328-346 (2001). MR 1814992 (2002b:46022)
- 2.
- D. Azagra, R. Deville and M. Jiménez-Sevilla, On the range of the derivatives of a smooth mapping between Banach spaces, Math. Proc. Cambridge Phil. Soc. 134 (2003), no. 1, 163-185. MR 1937801 (2004c:46069)
- 3.
- D. Azagra, M. Fabian and M. Jiménez-Sevilla, Exact filling in figures by the derivatives of smooth mappings between Banach spaces, Canadian Mathematical Bulletin, to appear.
- 4.
- D. Azagra and M. Jiménez-Sevilla, Geometrical and topological properties of starlike bodies and bumps in Banach spaces, Extracta Math. 17 (2002), no. 2, 151-200. MR 1937297 (2003j:46013)
- 5.
- D. Azagra and M. Jiménez-Sevilla, On the size of the sets of gradients of bump mappings and starlike bodies on the Hilbert space, Bull. Soc. Math. France 130 (2002), no. 3, 337-347. MR 1943881 (2003k:46056)
- 6.
- S. M. Bates, On smooth non-linear surjections of Banach spaces, Israel J. Math. 100 (1997), 209-220. MR 1469111 (98i:58016)
- 7.
- J.M. Borwein, M. Fabian, I. Kortezov, P.D. Loewen, The range of the gradient of a continuously differentiable bump, J. Nonlinear and Convex Anal. 2 (2001), 1-19. MR 1828155 (2002c:58012)
- 8.
- J.M. Borwein, M. Fabian, P.D. Loewen, The range of the gradient of a Lipschitz
 -smooth bump in infinite dimensions, Israel J. Math. 132 (2002), 239-251. MR 1952623 (2003j:58011) - 9.
- K. Kuratowski, Topologie I, PWN Warszawa 1966.
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- T. Gaspari, On the range of the derivative of a real valued function with bounded support, Studia Math. 153 (2002), no. 1, 81-99. MR 1948929 (2003k:46057)
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Additional Information:
Marián
Fabian
Affiliation:
Mathematical Institute, Czech Academy of Sciences, Zitná 25, 115 67 Praha 1, Czech Republic
Email:
fabian@math.cas.cz
Ondrej
F. K.
Kalenda
Affiliation:
Department of Mathematical Analysis, Faculty of Mathematics and Physics, Charles University, Sokolovská 83, 186 75 Praha 8, Czech Republic
Email:
kalenda@karlin.mff.cuni.cz
Jan
Kolár
Affiliation:
Department of Mathematical Analysis, Faculty of Mathematics and Physics, Charles University, Sokolovská 83, 186 75 Praha 8, Czech Republic
Email:
kolar@karlin.mff.cuni.cz
DOI:
10.1090/S0002-9939-04-07730-5
PII:
S 0002-9939(04)07730-5
Keywords:
$C^1$-smooth bump,
separable dual Banach space,
analytic set
Received by editor(s):
March 21, 2002
Posted:
August 24, 2004
Additional Notes:
The first author's research was supported by grants A101 90 03, A101 93 01 and GA CR 201/01/1198.
The second author's research was supported by grants GAUK 277/2001, MSM 113200007 and GA CR 201/00/1466
The third author's research was supported by grants MSM 113200007 and GA CR 201/02/D111
Communicated by:
Jonathan M. Borwein
Copyright of article:
Copyright
2004,
American Mathematical Society
The copyright for this article reverts to public domain after 28 years from publication.
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