Proving Corollary to Hahn-Banach Theorem: The Uniqueness of the Zero Point

[Zoe-b]

Homework Statement

To clarify- this isn't a homework problem; its something that's stated as a corollary in my notes (as in the proof is supposed to be obvious) and I haven't yet managed to prove it- I'm probably just missing something! Would appreciate a hint or a link to where I might find the proof.

Let X be a normed vector space, t $\in$ X such that for all g $\in$ X', g(t) = 0. Then t = 0.

Homework Equations

Hahn-Banach Theorem (stated on my course as:)
Let M be a subspace of a normed vector space X. Let f $\in$ M' . Then there exists g $\in$ X' such that the norm of f (wrt M') is equal to the norm of g (wrt X'), and g is equal to f on M.

The Attempt at a Solution

Just a bit confused- this is equivalent to, the only point all linear functionals can vanish at is zero. Presumably I want to define some linear subspace to then use but the only one that jumps out at me is span(t) which then doesn't seem to give me anything. Any hints?

 

[micromass]

You're going in the right direction. Assume that $t\neq 0$. Then set $M= span(t)$. Now define a suitable nonzero functional on $M$ and extend it by Hahn-Banach.

 

[Zoe-b]

Possibly in the right direction, but unfortunately not at speed :P

I want to find a linear functional f defined on M s.t. f vanishes at t (so that its Hahn-Banach extension will satisfy the given property).. but if f vanishes at t then it vanishes on the whole of M, which is not particularly useful. I'm probably being incredibly slow but would appreciate a further hint, sorry! Also I have a lot of exams to revise for and don't really want to spend loads of time trying to prove this one little bit... Thank you in advance for any help!

 

[micromass]

You want to assume that $t\neq 0$ and you want to find a linear functional that does not vanish at $t$. This would be a contradiction with the property that $t$ has (namely, that all the linear functionals vanish there.

So, can you find a linear functional $f:M\rightarrow \mathbb{R}$ with $M=span(t)$ such that $f(t)\neq 0$?

 

[Zoe-b]

Hmmn I think I was getting confused with the logic of what I was trying to do then. I can take f(at) = a |t| (where |t| is its norm). Then if every functional vanishes at t then the extension of f, g satisfies g(t) = 0 = |t| and by positive definiteness t is zero.