Norm of a functional I'm facing the problem of calculating the norm of the following functional:
$\displaystyle \phi : L_p([0,1]) \rightarrow \mathbb{R}, ~~ \phi (f) = \int\limits_{0}^{1} e^x f(x) dx  $
I have no idea where to begin. I need to see the supremum of $\displaystyle \phi $ on the set $ B(0,1)= \{f \in L_p([0,1]): \displaystyle\int_0^1|f(x)|^p dx \leq 1\} $

for every $f\in B(0,1)$ we have 
$$|\phi(f)|\leq \left(\int_0^1e^{qx}dx\right)^{1/q}||f||_p$$ where $1/p+1/q=1$, so $||\phi||\leq \frac{1}{q}(e^q-1)^{\frac1q}$
