CMS/SMC
Canadian Mathematical Society
www.cms.math.ca
Canadian Mathematical Society
  location:  PublicationsjournalsCMB
Publications        
Abstract view

Exact Filling of Figures with the Derivatives of Smooth Mappings Between Banach Spaces

  Published:2005-12-01
 Printed: Dec 2005
  • D. Azagra
  • M. Fabian
  • M. Jiménez-Sevilla
Features coming soon:
Citations   (via CrossRef) Tools: Search Google Scholar:
Format:   HTML   LaTeX   MathJax   PDF   PostScript  

Abstract

We establish sufficient conditions on the shape of a set $A$ included in the space $\mathcal L _s^n(X,Y)$ of the $n$-linear symmetric mappings between Banach spaces $X$ and $Y$, to ensure the existence of a $C^n$\nobreakdash-smooth mapping $f\colon X \rightarrow Y$, with bounded support, and such that $f^{(n)}(X)=A$, provided that $X$ admits a $C^{n}$-smooth bump with bounded $n$-th derivative and $\dens X=\dens \mathcal L ^n(X,Y)$. For instance, when $X$ is infinite-dimensional, every bounded connected and open set $U$ containing the origin is the range of the $n$-th derivative of such a mapping. The same holds true for the closure of $U$, provided that every point in the boundary of $U$ is the end point of a path within $U$. In the finite-dimensional case, more restrictive conditions are required. We also study the Fr\'echet smooth case for mappings from $\mathbb R^n$ to a separable infinite-dimensional Banach space and the G\^ateaux smooth case for mappings defined on a separable infinite-dimensional Banach space and with values in a separable Banach space.
MSC Classifications: 46B20 show english descriptions Geometry and structure of normed linear spaces 46B20 - Geometry and structure of normed linear spaces
 

© Canadian Mathematical Society, 2014 : http://www.cms.math.ca/