Hahn Banach extension of linear functional $f$

Question

Let $f:(c_{00},\|\cdot\|_1)\to \mathbb C $ be a non zero continuous linear functional. The number of Hahn-Banach extensions of f to $(\ell^1,\|\cdot\|_1)$ is

1. one
2. two
3. infinite
4. three

I have no idea how to get the result.

Hahn Banach theorem on normed linear space: If I have a linear functional defined on some subspace then I can extend it on the whole vector Space with same norms.

Here $\|x\|_1= \sum_{n=1}^\infty |x_n|$,where $x=(x_1,x_2....)\in c_{00} = $ the set of all sequences whose finite terms are non-zero.

If $x\in \ell^1$, then $\|x\|_1= \sum_{n=1}^\infty |x_n|<\infty $.

Someone help. Thanks.

Answer 1

Let my give some hints which should help to solve the problem:

• If we have two different extensions, all convex combinations are still extensions, hence, the answer is either $1$ or $\infty$.
• $c_{00}$ is dense in $\ell^1$. And we get only unique extension if the subspace is dense in X