Hopefully easy question about sups of continuous functionsby AxiomOfChoice Tags: continuous, functions, sups 

#1
Mar2211, 11:12 AM

P: 517

If f is a continuous functional on a normed space, do you have
[tex] \sup_{\x\ < 1} f(x) = \sup_{\x\ = 1} f(x) [/tex] If so, why? If not, can someone provide a counterexample? 



#2
Mar2211, 07:59 PM

Sci Advisor
P: 3,172

That's an interesting question. A normed space must be a vector space right? But does a vector space have to be connected?




#3
Mar2211, 11:29 PM

Sci Advisor
HW Helper
P: 9,421

try it on R. can you think of a function which gets larger somewhere inside the unit interval than it is at 1, 1? maybe you meant linear. or maybe functional means linear. then it seems true.




#4
Mar2311, 10:36 AM

P: 517

Hopefully easy question about sups of continuous functionsAnd yeah, it clearly seems true on R, but for an arbitrary normed space, I'm not so sure... 


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