1. **Problem statement:** We want to prove the duality relation between $L^p$ and $L^q$ spaces for $1 < p < \infty$, where $\frac{1}{p} + \frac{1}{q} = 1$. Specifically, for $f \in L^p(\Omega, A, \mu)$, we want to show $$ ||f||_p = \sup \left\{ \left| \int f g \, d\mu \right| \mid g \in L^q(\Omega, A, \mu), ||g||_q \leq 1 \right\}. $$ 2. **Recall Hölder's inequality:** For $f \in L^p$ and $g \in L^q$, $$ \left| \int f g \, d\mu \right| \leq ||f||_p ||g||_q. $$ This implies $$ \sup_{||g||_q \leq 1} \left| \int f g \, d\mu \right| \leq ||f||_p. $$ 3. **Constructing a function $h$ to achieve equality:** Define $$ h := f |f|^{p-2} 1_{\{f \neq 0\}}. $$ Note that $q(p-1) = p$. 4. **Calculate the $L^q$ norm of $h$:** $$ ||h||_q^q = \int |h|^q d\mu = \int |f|^{q(p-1)} d\mu = \int |f|^p d\mu = ||f||_p^p. $$ 5. **Calculate the inner product $\langle f, h \rangle$:** $$ \langle f, h \rangle = \int f \overline{h} \, d\mu = \int |f|^p d\mu = ||f||_p^p. $$ 6. **Normalize $h$ to get $g$ with $||g||_q = 1$:** $$ g := \frac{h}{||h||_q} = \frac{h}{||f||_p^{p/q}} = \frac{h}{||f||_p^{p-1}}. $$ 7. **Express $||f||_p$ in terms of $g$:** $$ ||f||_p = \frac{||f||_p^p}{||f||_p^{p-1}} = \langle f, g \rangle = \int f \overline{g} \, d\mu. $$ 8. **Conclusion:** We have shown $$ ||f||_p \leq \sup_{||g||_q \leq 1} \left| \int f g \, d\mu \right| \leq ||f||_p, $$ so equality holds: $$ ||f||_p = \sup_{||g||_q \leq 1} \left| \int f g \, d\mu \right|. $$ 9. **Extension to the mapping $\iota$: ** For $f \in L^q(\Omega, A, \mu)$, define $$ \iota(f) := \langle f, \cdot \rangle : L^q(\Omega, A, \mu) \to \mathbb{K}, $$ which is linear and continuous by Hölder's inequality, establishing the isometric isomorphism between $L^p$ and the dual space of $L^q$. This completes the proof of the duality and norm characterization.