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\} $

$$\left|\int_0^1e^xf(x)dx\right|\stackrel{\text{ Hoelder's ineq.}}\leq \left|\int_0^1e^{qx}dx\right|^{1/q}\|f\|_p=\frac1q(e^q-1)^{1/q}\|f\|_p$$
The supremum can be attained in this case. Try out $f(x)=1$.
