I have some trouble solving this problem, can anyone help? For $g\in C[a,b]$ define $\| g\|_{1} =\int\limits_a^b |g(t)|dt$ and consider $\ell:C[a,b] \to \mathbb{F}$ given by $\ell(g)=\int_a^b g(t)dt$.
I need prove that the linear funcional $\ell$ is bounded and I have to compute $\|\ell\|$. I think this is solveable, but now I have to prove that there exist a unique bounded linear functional $\bar{\ell}:L^{1}[a,b] \to \mathbb{F}$ that extends $\ell$ to the $\|\cdot\|_{1}$ completion $L^{1}[a,b]$
I think that I have to use that for every $1\leq p < \infty$ the space $L^{p}[a,b]$ is a completion of $C[a,b]$ with respect to the norm $\| g\|_{p} =(\int\limits_a^b |g(t)|^{p}dt)^{1/p}$, but I don't know how exactly. This seems crazy, any ideas?
We have $$|\ell(g)|=\left|\int_a^bg(t)dt\right|\le\int_a^b|g(t)|dt=\|g\|_1,$$ so $\ell$ is bounded with $\|\ell\|\le1$. If $g\ge0$ (for example $g(t)=1$), then in fact $\ell(g)=\|g\|_1$ so we must have $\|\ell\|=1$.
The second part of your question comes from the density of $C[a,b]$ in $L^1(a,b)$ and a well-known result. I would suggest attempting to prove this theorem yourself, but there is surely a proof somewhere on this site.