Let $X = C([0,1])$, and $L = Span(t)$. Defining the functional $f(x):= \lambda$ when $x \in L$ is $\lambda t$.
Clearly, $||f||_{L^*} = 1$.
From Hahn-Banach, it can be extended to $F \in X^*$, with $||F||_{X^*} = 1$.
The functional $\phi$ defined by $\phi(z) = z(1)$ works as a norm-preserving extension.
My question is, is this extension unique, or is there another one? In case it is unique, how would one go on about proving it?
EDIT: By $t$ I mean the identity in $[0,1]$