V* ≠ {0}

Fredrik

Is there an easy way to see that if V is a (non-empty and non-trivial) normed space, there exists a non-zero continuous linear functional on V? To put it another way: Is there an easy way to see that V*≠{0}?. Do we have to use the Hahn-Banach theorem for this?

micromass

If V is finite dimensional, then V* is always nonempty.
But for infinite-dimensional space, you will have to use Hahn-Banach. This theorem will give you a nontrivial bounded functional.

CompuChip

Sorry if this is nonsense, it's been a while, but isn't the norm continuous by definition?
I.e. isn't
[tex]|| \cdot || : V \to \mathbb{R}, f \mapsto ||f||[/tex]
a linear functional on V?
Clearly it is non-zero, because ||f|| = 0 iff f = 0.

arkajad

Norm is not linear because for [itex]v\neq 0[/itex]

[tex]||-v||\neq -||v||[/tex]

CompuChip

Good point.
Never mind I said something.

* whistles and shuffles away innocently *

Fredrik

Thanks, that's what I thought. I got a little confused by the fact that immediately before stating the Hahn-Banach theorem, the book suggested that I think about why V*≠{0}. It's hard to know if I'm supposed to try and succeed, or try and fail just to see that we need a fancy theorem.