Advanced Search

Query:

Fill in the form and click »Search«...

Format:

Display: entries per page entries

Zbl 1056.46032
Araujo, Jesús; Jarosz, Krzysztof
Automatic continuity of biseparating maps. (English)
[J] Stud. Math. 155, No. 3, 231-239 (2003). ISSN 0039-3223; ISSN 1730-6337/e

Let $X$, $Y$ be realcompact spaces, $E$ and $F$ normed spaces, $C(X,E)$ the set of all continuous $E$-valued functions on $X$ and $C_b(X,E)$ the space of all bounded continuous functions from $X$ into $E$. A linear map $T: C(X,E) \to C(Y,F)$ is called separating if $\Vert f(x)\Vert \ \Vert g(x)\Vert =0$ for every $x\in E$ implies that $\Vert (Tf)(y)\Vert \ \Vert (Tg)(y)\Vert =0$ for every $y\in Y$, and biseparating if $T^{-1}$ exists and is separating as well. One result of the paper is that every linear biseparating map $T:C_b(X,E)\to C_b(Y,F)$ is continuous provided that $Y$ has no isolated points. Another result tells us that if additionally $E$ and $F$ are infinite-dimensional, then a linear biseparating map $T: C(X,E)\to C(Y,F)$ is continuous if the interior of the set of $P$-points of $Y$ is empty. This is the best possible result.
[Raymond Mortini (Metz)]

MSC 2000:: *46E40 Spaces of vector-valued functions
47B33 Composition operators
46H40 Automatic continuity
47B38 Operators on function spaces
46E25 Rings and algebras of functions with smoothness properties

Keywords: automatic continuity of biseparating maps; realcompact spaces; P-points

Comment on this Item PDF MathML XML ASCII DVI PS BibTeX Online Ordering Article Journal Journal

	Scientific prize winners of the ICM 2010
	Overhang
	Lie groups, physics and geometry. An introduction for physicists, engineers and chemists.

Advanced Search

Zentralblatt MATH Berlin [Germany]