Let $p \in [1,\infty)$ and let $\Lambda$ be a linear functional on $L^P(I)$ denoted by
$$\Lambda(f) = \int_0^1 e^{2x} f(x) dx \;\;\; \text{for } f\in L^P(I).$$
where $I = [0,1] \subset \mathbb{R}$. Use $\|f\|_{p}^p = \int_0^1 |f(x)|^pdx$ for $f \in L^P(I)$.
I show that
\begin{align*} |\Lambda f| &= \bigg| \int_0^1 e^{2x} f(x) dx \bigg | \\ & \leq \int_0^1 |e^{2x} f(x)| dx \\ & \leq \|e^{2x}\|_q \|f(x)\|_p < \infty \end{align*}
Therefore $\| \Lambda\| \leq \|e^{2x}\|_q$. I think I need to show that $\| \Lambda \| \geq \| e^{2x} \|_q$ to get norm of this linear functional. But I couldn't find a way to get that inequality. How could I find the norm of $\Lambda$.
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By using the condition for equiality in Holder's inequality, let $f(x) = e^{2xq/p}$.
Then $| \Lambda f | = \|e^{2x}\|_q$. Therefore $\| \Lambda \| = \|e^{2x}\|_q$.