Mentor

## V* ≠ {0}

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?
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 Blog Entries: 8 Recognitions: Gold Member Science Advisor Staff Emeritus 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.
 Blog Entries: 5 Recognitions: Homework Help Science Advisor 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.

## V* ≠ {0}

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

$$||-v||\neq -||v||$$
 Blog Entries: 5 Recognitions: Homework Help Science Advisor Good point. Never mind I said something. * whistles and shuffles away innocently *
 Mentor 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.