Hahn Banach Theorem

Question

It is stated often that the Hahn Banach Theorem makes the study of the dual space "interesting". What does this exactly mean though?

I.e what is exactly meant by "interesting"?

I am puzzled as to why it follows immediately from Hahn-Banach that the dual of a (non-zero) normed vector space is non-trivial.

How does it follow DIRECTLY from Hahn Banach that there are non-trivial functions?

A note on what you wrote: the only constant function in the dual is the zero function. Remember that the elements of the dual are linear functions. — Daniel Robert-Nicoud, Mar 17 '13 at 23:04

Lepidopterist · Accepted Answer · 2013-03-17T23:14:59.390

17

A consequence of Hahn Banach is that linear functionals separate points. This implies a certain richness of the space of linear functionals.

Separating points means that given two distinct points $x$ and $y$ there is a continuous linear functional $f$ such that $f(x)\neq f(y)$.

Added in light of your inquiry: to prove that there is such a functional, consider the one-dimensional subspace $\mathbb{C}(x-y)$ (complex multiples of $x-y$). You can easily show that on this subspace $f(\lambda (x-y))=\lambda||x-y||$ defines a continuous linear functional. You can then extend this to your whole space by Hahn-Banach and by linearity it will follow that $f(x)-f(y)=f(x-y)=||x-y||\neq 0$, so $f(x)\neq f(y)$, as desired.

edited Mar 17 '13 at 23:14

answered Mar 17 '13 at 22:54

Lepidopterist

2,751

Thank you for your reply. How does this follow directly from Hahn-Banach though? (I'm thinking of the theorem that guarantees extensions of linear functionals on subspaces to the whole space) – Mar 17 '13 at 22:56
1

Thank you! Is this the usual argument to show the dual is non-trivial? – Mar 17 '13 at 23:18
Yes, I believe so. – Lepidopterist Mar 17 '13 at 23:19

Hahn Banach Theorem

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