MathOverflow will be down for maintenance for approximately 3 hours, starting Monday evening (06/24/2013) at approximately 9:00 PM Eastern time (UTC-4).
0

Is $C^1[0,1]$ linearly homeomorphic to $C[0,1]$ ?

flag
1 
 Mr Smith, you definitely have Slavic roots: "gomeomorphic" gives yourself completely... Can't you put more effort on your question? It sounds a homework. – Wadim Zudilin Jan 8 2011 at 12:48
1 
 Dear Mr Smith, Wadim was merely suggesting what you could have already read here, mathoverflow.net/howtoask – Gjergji Zaimi Jan 8 2011 at 13:07
1 
 Mr Smith, I have merged your two user accounts. If you wish to respond to someone else's answer, please leave a comment instead of creating a new answer. Please try to be civil. – S. Carnahan♦ Jan 8 2011 at 13:30

closed as too localized by Andres Caicedo, Yemon Choi, Harry Gindi, Wadim Zudilin, Bill Johnson Jan 11 2011 at 18:47

1 Answer

4

By integration, $C[0,1]$ is linearly homeomorphic to a hyperplane in $C^1[0,1]$. For example, all elements $g$ of $C^1[0,1]$ with $g(0)=0$. All classical Banach spaces are linearly isomorphic to their hyperplanes. In this case it can be proved constructively.

So, Mr Smith, one-visit wonder: Is this your homework?

link|flag
8 
 "It is believed that all Banach spaces are linearly isomorphic to their hyperplanes."? Banach's Hyperplane Problem was solved into negative by W.T.Gowers in 1993. – Ady Jan 8 2011 at 14:38
thanks, edited. – Gerald Edgar Jan 8 2011 at 16:53

Not the answer you're looking for? Browse other questions tagged or ask your own question.