Extending linear functional in non-unique way

Question

I'm trying to find an example of when the extension of a functional in the Hahn-Banach theorem is not necessarily unique. I'm looking at the space of continuous functions on $[0,1]$ and I'm trying to find a subspace and a functional defined on the subspace that has two different norm preserving extensions.

So far I've thought about the subspace of polynomials, but then the only linear functionals I can think of are things like $f(p) = p(0)$ or $f(p) = \|p(x)\|_{\infty}$ but I can't think of any non-unique way to extend these.

Answer 1

What about the subspace $$S=\{f\in C(0,1); f(0)=f(1)\}$$ and the functional $T\colon f \mapsto f(0)$ defined on $S$.

The you have two extensions $T_0\colon f\mapsto f(0)$ and $T_1\colon f\mapsto f(1)$.

For these functionals we have $\|T\|=\|T_1\|=\|T_2\|=1$.

In fact, for every $a\in[0,1]$ we have an extension $T_a\colon f\mapsto af(0)+(1-a)f(1)$ with $\|T_a\|=1$.

Answer 2

The following is true:

If $X$ is a Banach space, then each Hahn-Banach (norm preserving) extension is unique iff the unit ball of $X^{\ast}$ is strictly convex.

In particular, this would be true for a Hilbert space. Furthermore, there is an interesting theorem of Phelps (See this) which relates this property for a fixed subspace to a best approximation property of the annihilator of that subspace in $X^{\ast}$.