Let $L$ be the linear subspace of $C[0,1]$ given by $L = \{x(t) \in C[0,1] : x(t) = \text{constant}\}$
Let $f_0\in L^*$, $f_0(x)= x(0)$ and $L_1=\text{span}(L,t)$ . Find a functional $f\in L_1^*$ such that $f|_c=f_0$ and $||f||=||f_0||$
Please provide me some hints/answer
I think that $f|_c$ means $f|_L$. Furthermore I suppose that $C[0,1]$ is eqipped with the usual $|| \cdot||_{\infty} - $ norm.
We have: $x \in L_1 \iff $ there are $x_1 \in L$ and $a \in \mathbb R$ such that $x(t)=x_1(t)+at.$
Now define $f\in L_1^*$ by $f(x):=x(0).$
For $x \in L$ we have $a=0,$ hence $x=x_1$ and therefore $f(x)=f_0(x).$
This shows that $$f|_L=f_0.$$
It is your turn to show that $||f_0||=1=||f||.$