Number of Hahn-Banach Extensions

Question

Let $f : (c_{00},\|\cdot\|_1)\rightarrow \mathbb C$ be a non-zero continuous linear functional. The number of Hahn-Banach extensions of $f$ to $(c,\|\cdot\|_1)$ is, I think, one. Here, $c_{00}$ is the space of eventually null sequences in $\mathbb C$, $c$ is the space of convergent sequences in $\mathbb C$ and $\|\cdot\|_1$ is the $\sum_{n=0}^{\infty}|x_{n}|$ norm. Is it true, because I think $c_{00}$ is dense in $c$ with the given norm.

Answer 1

First, you should equip $c$ with the sup-norm. As pointed out, the harmonic sequence is in $c$ but does not have finite $\|\cdot\|_1$ norm.

Also, $c_{00}$ is not dense in $c$. Take for instance $x=(1,1,1,\dots)\in c$. Then $\|x-y\|_\infty\ge 1$ for all $y\in c_{00}$.

The limit functional, $\lim:c\to \mathbb C$ is linear and continuous. It is non-zero, of course, but its restriction to $c_{00}$ is zero. Thus, you can add a multiple of the limit functional to your extension of $f$ to obtain another, different, extension. Hence the extension is not unique.

The extension of a densely defined and bounded functional is unique. Hence, the extension to $c_0$ is unique.