Hopefully easy question about sups of continuous functions

[AxiomOfChoice]

If f is a continuous functional on a normed space, do you have

$$\sup_{\|x\| < 1} |f(x)| = \sup_{\|x\| = 1} |f(x)|$$

If so, why? If not, can someone provide a counterexample?

 

[Stephen Tashi]

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

 

[mathwonk]

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.

 

[AxiomOfChoice]

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.

Yeah, I did mean to put "linear functional" above...I guess I was under the impression that "functional" implies "linear functional", though I'm not at all sure that's the case.

And yeah, it clearly seems true on R, but for an arbitrary normed space, I'm not so sure...