Re: continuos linear image of open sets

From: Jannick Asmus (jannick.news_at_web.de)
Date: 03/24/05 

Date: Thu, 24 Mar 2005 16:00:11 +0000 (UTC)

On 24.03.2005 03:30, Jose Gascon wrote:
> It is well known that if T:X->X, X Banach space, is a continuos linear
> operator and surjective T maps open sets to open sets. What about if
> you drop the hypothesis that T is surjective, is T(0), O open in X,
> Borel?
> Thanks for any help(maybe this is well known)
> Jos,bi(B Gasc,bs(Bn
>

I think your assertion is right, if in addition you assume X to be
separable.

I hope that I remember the following theorem correctly: An injective
measurable map f: A -> B, A,B complete separable metric spaces, maps
Borel subsets of A to such of B. This should be found in

@BOOK{Parthasarathy1967,
   title = {Probability measures on metric spaces},
   publisher = {Academic Press},
   year = {1967},
   author = {Kalyanapuram R. Parthasarathy},
}

J.

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