I'm very recent to functional analysis. My homework problem states: Let $x_0(t) \in C[0, 1]$ a fixed continuous function and $ L = span(x_0) $. Consider the functional in $L$ defined as:
$$ f(x) := \lambda \quad \text{iff} \quad x = \lambda x_0$$
a) Prove $ |f|_{L*} = 1 $
b) According to Hahn Banach, there is an extension $F \in X*$ such that $|F|_{C*} = 1$ . Is it unique when considering: $$ x_0(t) = t $$ And $$ x_0(t) = 1-2t $$
Part a) is quite easy (plugging the definition) and I can see why the hypothesis of Hahn Banach are satisfied. Yet this only tells me I can indeed build an extension. However, I fail to see what the definition of an extension is given $x_0$ and I can't tell how the answer would even look like. I noticed that the first function attains its maximum (in absolute value) at $t= 1$ while the other one has $t=0$ and $t=1$.
Considering this, my intuition tells me, that perhaps the second function can be extended in more than one way. Yet I'm unable to tell an adequate answer. Please help.
More concretely, if the answer were for example "In the second function I can make at least two extensions..." Would I have to provide examples? If I say there is only one, I'd have to show any other would be the same as the first extension found. How do I do that??
This is my first question btw. Any advice would be truly appreciated.