Theorem. For $p\in [1, \infty)$, $L^p$ is complete.
Proof. Pick a Cauchy sequence $f_n$. We can pick a subsequence $f_{n_k}$ such that $\|f_{n_{k+1}}- f_{n_k}\| \leq 2^{-k}$. Set $S_K(f) = f_{n_1} + \sum_{i=1}^{K} (f_{n_{i+1}} – f_{n_i})$ and also $S_K(g) = |f_{n_1}| + \sum_{i=1}^{K} |f_{n_{i+1}}- f_{n_i}|$. By the triangle inequality on the $p$-norm, $\|S_K(g)\|_p \leq \|f_{n_1}\|_p + \sum_{i=1}^K \|f_{n_{i+1}}- f_{n_i}\|_p \leq \|f_{n_1}\|_p + 1$, so by Monotone Convergence, $S_\infty(g) \in L^p$, hence $S_\infty(f)$ is also in $L^p$, hence it converges almost everywhere to some $f\in L^p$. But clearly $S_K(f) = f_{n_{K+1}}$, so $f_{n_k} \to f$ almost everywhere. You can then show using DCT that $f_{n_k} \to f$ in the $p$-norm. By the Cauchy property of $f_n$, you then also show that $f_n \to f$ in $L^p$. $\blacksquare$
Theorem. Suppose $p, q \in [1, \infty]$ are conjugates, and $fg$ is integrable for each $g\in L^q$. Then $f\in L^p$.
Proof. We can find a sequence of simple functions $s_k$ which converges to $f$ in $L^1$ with $|s_k| \leq |f|$. We may choose these such that $|s_k(x)|$ is increasing for each $x\in X$. Each $s_k$ trivially belongs to $L^p$, by simplicity. Now, let $S_k(h) = \int s_kh$ be a functional on $L^q$. Linearity is obvious. Boundedness follows from Hölder’s Inequality, and in fact the extremal form thereof implies that $\|S_k\|_* = \|s_k\|_p$.
Pick any $h\in L^q$. Then DCT and our assumption that $fh\in L^1$ implies $\lim_k \int s_kh = \int \lim_k s_kh = \int fh \in \Bbb C$, hence $\sup_k \{ |S_k(h)| \} < \infty$. As $h$ was arbitrary, Uniform Boundedness implies that $\sup_k \{ \|S_k\|_*\} = \sup_k \{ \|s_k\|_p\}< \infty$. However, $\lim_k \int |s_k|^p = \int |f|^p$ by Monotone Convergence, so $f\in L^p$. $\blacksquare$
Theorem. Suppose $p\in (1, \infty)$ and let $D = (L^p)^*$, the dual of $L^p$. Then $D$ is isomorphic to $L^q$, where $1/p + 1/q = 1$. (We’re assuming an ambient $\sigma$-finite measure space $(X, \Sigma, \mu)$ here.)
Proof. D is of course already a Banach space with norm
$$\|T\|_D = \inf \{ M : |Tf| \leq M \|f\|_p \text{ for all } f\in L^p \}.$$
We want to map the bounded linear functional $T$ on $L^p$ to some function $g_T \in L^q$ such that $\|T\|_D = \|g_T\|_q.$
Let’s first show that $T\in D$ implies a $g$ such that $Tf = \int fg$. When $f$ is the characteristic function of some measurable set $A$, we can define a complex set function by $\nu(A) = Tf$. By linearity, $\nu$ is already finitely additive. On the other hand, take a countable disjoint collection $A_i$ and define $B_i = \bigcup_{k=i}^\infty A_i$; assume $1_{B_0}\in L^p$. Finite additivity implies that for each $k\in\Bbb N$,
$$\nu(B_0) = \nu(A_0) + \ldots + \nu(A_k) + \nu(B_{k+1})\tag{1}.$$
But $\nu(B_k) = T1_{B_k}$. Boundedness of $T$ implies
$$\begin{eqnarray}
|T1_{B_k}| &\leq& \|T\|_D \|1_{B_k}\|_p \\
&=& \|T\|_D \left( \int_{B_k} d\mu \right)^{1/p}.
\end{eqnarray}$$
Now, $B_k \to \emptyset$, and since $B_0 \supseteq B_k$ is of finite measure (as $1_{B_0} \in L^p$), $\int_{B_k} d\mu \to 0$. Therefore $\nu(B_k) \to 0$, so taking limits of $(1)$ establishes countable additivity of $\nu$. Finally, for any $\mu$-null set $A$, $|\nu(A)| = |T1_A| \leq \|T\|_D \left( \int_A d\mu \right)^{1/p} = 0$, hence $\nu \ll \mu$. So Radon-Nikodym implies that $T1_A = \nu(A) = \int_A g\ d\mu$. It is not difficult to see that this, together with continuity of $T$, implies $Tf = \int gf\ d\mu$ for $f\in L^p$. Our previous theorem then implies that $g\in L^q$.
Hölder’s Inequality thus implies that $|Tf|\leq \int |fg|\leq \|f\|_p \|g\|_q$, so that $\|T\|_D \leq \|g\|_q$. Hölder’s Inequality is also sharp, so we can select $f$ to make this an equality.