V* ≠ {0}: Exploring Non-Zero Continuous Linear Functionals in Normed Spaces

[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
$$|| \cdot || : V \to \mathbb{R}, f \mapsto ||f||$$
a linear functional on V?
Clearly it is non-zero, because ||f|| = 0 iff f = 0.

 

[arkajad]

Norm is not linear because for $v\neq 0$

$$||-v||\neq -||v||$$

 

[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.