$L^p$ spaces

Theorem. For $p\in [1,\mathrm{\infty })$, $L^p$ is complete.

Proof. Pick a Cauchy sequence $f_n$. We can pick a subsequence $f_{n_k}$ such that $\Vert f_{n_{k+1}}-f_{n_k}\Vert \le 2^{-k}$. 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, $\Vert S_K(g){\Vert }_p\le \Vert f_{n_1}{\Vert }_p+\sum _{i=1}^K\Vert f_{n_{i+1}}-f_{n_i}{\Vert }_p\le \Vert f_{n_1}{\Vert }_p+1$, so by Monotone Convergence, $S_{\mathrm{\infty }}(g)\in L^p$, hence $S_{\mathrm{\infty }}(f)$ 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 $L^p$. $\mathrm{<img\; draggable="false"\; class="emoji"\; alt="◼"\; src="https://s.w.org/images/core/emoji/11/svg/25fc.svg">}$

Theorem. Suppose $p,q\in [1,\mathrm{\infty }]$ 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 $|s_k|\le |f|$. 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 $S_k(h)=\int s_{k}h$ be a functional on $L^q$. Linearity is obvious. Boundedness follows from Hölder’s Inequality, and in fact the extremal form thereof implies that $\Vert S_k{\Vert }_{\ast }=\Vert s_k{\Vert }_p$.

Pick any $h\in L^q$. Then DCT and our assumption that $fh\in L^1$ implies $\underset{k}{lim}\int s_{k}h=\int \underset{k}{lim}{s}_{k}h=\int fh\in \mathbb{C}$, hence $\underset{k}{sup}\{|S_k(h)|\}<\mathrm{\infty }$. As $h$ was arbitrary, Uniform Boundedness implies that $\underset{k}{sup}\{\Vert S_k{\Vert }_{\ast }\}=\underset{k}{sup}\{\Vert s_k{\Vert }_p\}<\mathrm{\infty }$. However, $\underset{k}{lim}\int |s_k{|}^p=\int |f{|}^p$ by Monotone Convergence, so $f\in L^p$. $\mathrm{<img\; draggable="false"\; class="emoji"\; alt="◼"\; src="https://s.w.org/images/core/emoji/11/svg/25fc.svg">}$

Theorem. Suppose $p\in (1,\mathrm{\infty })$ and let $D=(L^p{)}^{\ast }$, 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 $(X,\mathrm{\Sigma },\mu )$ here.)

Proof. D is of course already a Banach space with norm

$$\Vert T{\Vert }_D=inf\{M:|Tf|\le M\Vert f{\Vert }_p\text{ for all }f\in L^p\}.$$

We want to map the bounded linear functional $T$ on $L^p$ to some function $g_T\in L^q$ such that $\Vert T{\Vert }_D=\Vert g_T{\Vert }_q.$ 

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 $B_i=\bigcup _{k=i}^{\mathrm{\infty }}{A}_i$; assume $1_{B_0}\in L^p$. Finite additivity implies that for each $k\in \mathbb{N}$,

$$\begin{array}{}\text{(1)}& \nu (B_0)=\nu (A_0)+\dots +\nu (A_k)+\nu (B_{k+1}).\end{array}$$

But $\nu (B_k)=T{1}_{B_k}$. Boundedness of $T$ implies

$$\begin{array}{rcl}|T{1}_{B_k}|& \le & \Vert T{\Vert }_D\Vert 1_{B_k}{\Vert }_p\\ & =& \Vert T{\Vert }_D{\left({\int }_{B_k}d\mu \right)}^{1/p}.\end{array}$$

Now, $B_k\to \mathrm{\varnothing }$, 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 $A$, $|\nu (A)|=|T{1}_A|\le \Vert T{\Vert }_D{\left({\int }_{A}d\mu \right)}^{1/p}=0$, hence $\nu \ll \mu$. So Radon-Nikodym implies that $T{1}_A=\nu (A)={\int }_{A}gd\mu$. It is not difficult to see that this, together with continuity of $T$, implies $Tf=\int gfd\mu$ for $f\in L^p$. Our previous theorem then implies that $g\in L^q$.

Hölder’s Inequality thus implies that $|Tf|\le \int |fg|\le \Vert f{\Vert }_p\Vert g{\Vert }_q$, so that $\Vert T{\Vert }_D\le \Vert g{\Vert }_q$. Hölder’s Inequality is also sharp, so we can select $f$ to make this an equality.