Theorem (Minkowski’s Inequality). If $f, g \in L^p$ for $1 \leq p < \infty$, $\|f + g\|_p \leq \|f\|_p + \|g\|_p$.
Proof. Start by noticing that $|f + g|^p \leq 2^{p-1}(|f|^p + |g|^q)$ by Jensen’s inequality, so that $\|f + g\|_p$ exists and is finite.
Let $1/p + 1/q = 1$, which is equivalent to $p + q = pq$. Notice that
$$
\begin{eqnarray}
\|(f+g)^{p-1}\|_q &=& \left( \int \left(|f + g|^{p-1}\right)^q d\mu \right)^{1/q} \\
&=& \left( \int |f + g|^{pq-q} d\mu \right)^{1/q} \\
&=& \left( \int |f + g|^p d\mu \right)^{1/q} \\
&=& \| f+g \|_p^{p/q}
\end{eqnarray}
$$
Now, by Hölder’s inequality, we see that $\| f(f+g)^{p-1} \|_1 \leq \|f\|_p\cdot \|(f+g)^{p-1}\|_q$, and similarly $\| g(f+g)^{p-1} \|_1 \leq \|g\|_p\cdot \|(f+g)^{p-1}\|_q$. Ergo
$$
\begin{eqnarray}
(\|f\|_p + \|g\|_p)\cdot \|(f+g)\|_p^{p/q} &=& (\|f\|_p + \|g\|_p) \cdot \|(f+g)^{p-1}\|_q \\
&\geq& \| f(f+g)^{p-1} \|_1 + \| g(f+g)^{p-1} \|_1 \\
&=& \int (|f| + |g|)|f + g|^{p-1} d\mu \\
&\geq& \int |f + g| \cdot |f + g|^{p-1} d\mu \\
&=& \int |f + g|^p d\mu \\
&=& \| f+g \|_p^p.
\end{eqnarray}
$$
Dividing the leftmost expression through by $\|(f+g)\|_p^{p/q}$ gives us one side of Minkowski’s inequality. Dividing through the rightmost gives us $\| f+g \|_p^{p – p/q} = \| f+g \|_p$, which is the other. $\blacksquare$
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 $L^p$.
Theorem. Let $B$ be a Banach space. Then its dual $C$ is also a Banach space.
Proof. Let $x \in B$ and $T, S \in C$; we define $(T + S)(x) = Tx + Sx, (\alpha T)x = \alpha(Tx)$. We define $\| T \|_C = \inf\ \{ M :|Tx| \leq M \|x\|_B \text{ for all } x \in B \}$. (We used $|Tx|$ because this is a scalar.) This of course implies $|Tx| \leq \|T\|_C \|x\|_B$. This norm obeys the triangle inequality because for all $x\in B$,
$$
\begin{eqnarray}
|(T + S)x| &=& |Tx + Sx| \\
&\leq& |Tx| + | Sx| \\
&\leq& \|T\|_C \|x\|_B + \|S\|_C \|x\|_B \\
&=& (\|T\|_C + \|S\|_C) \|x\|_B
\end{eqnarray}
$$
and $\|T+S\|_C$ is the infimum of all constants with this property.
Suppose $T_n\in C$ is a Cauchy sequence of bounded linear functionals on $B$. For all $x\in B$, our definitions imply that $|T_n x- T_m x| \leq \|T_n – T_m\|_C \| x\|_B$, so since the first factor can be made arbitrarily small for sufficiently large indices, it follows that $T_n x$ is a Cauchy sequence on $B$’s scalar field, hence it converges to some scalar we’ll call $T x$. Thus we’ve shown $T_n$ converges to a functional $T$. Clearly $T$ is linear.
We need to show that $T$ is bounded. Note that by the reverse triangle inequality, $\|T_n – T_m\|_C \geq |\|T_n\|_C- \|T_m\|_C|$. Since the left side can be made arbitrarily small for all large indices by our original assumption, so can the right side, hence $\|T_n\|_C$ forms a Cauchy sequence, so it converges to some constant. It follows, therefore, that $\|T_n\|_C$ are all uniformly bounded by some $M$. So $|Tx| \leq \|T – T_n\|_C \|x\|_B + \|T_n\|_C \|x\|_B$. Convergence of $T_n$ to $T$ in the $C$-norm implies the left term can be made arbitrarily small regardless of $x$ by appropriate choice of $n$, and the right term is bounded by $M\|x\|_B$. It follows that $\|T\|_C \leq M$ and so $T$ is bounded.