Hi,
I've problem with following statement:
For (real Banach algebra) the functional
satisfy
Any help or advices will be highly appreciated,
Best regars
Hi,
Below is link to the paper where is that statement. Remark after proof of second Theorem.
I didn't see that mistake befor, of course it should be
.
And You are right it is algebra of all continuous functions on the unit interval.
I'm sorry for that mistake.
PS. Here is link to the original paper http://matwbn.icm.edu.pl/ksiazki/sm/sm29/sm29126.pdf
Here the Banach algebra of all real valued bounded continuous functions on . For let . Then one easily proves that . Though the infimum and the supremum may not be attained, one can choose in the range of such that . Now applying the Intermediate Value Theorem, one concludes that there is some such that is in the range of . Consequently, is not a continuous function, in fact it is not defined at . Thus . Since is chosen arbitrary, we see that the result is true for all