What is $L^p$-convergence useful for?

It's difficult to know what you find interesting without further information. But here's an application from partial differential equations, anyway.

Many undergraduates learn how to solve the initial value problem $$ \begin{cases} u_t + u_{xxx} = 0 \qquad x,t\in\mathbb{R}\\ u(x,0) = u_0(x). \end{cases} $$ Using the Fourier transform, the solution is written $$ u(x,t) = U(t)u_0(x) = [e^{it\xi^3}\hat{u}_0]^\vee(x). $$ A basic calculus class will usually provide a description of the solution procedure, side stepping convergence issues. What are these issues?

For concreteness, suppose $u_0 \in L^2(\mathbb{R})$. Since $|e^{it\xi^3}|=1$, two applications of Parseval's theorem shows $$ \|U(t)u_0\|_2 = \|e^{it\xi^3}\hat{u}_0\|_2 = \|\hat{u}_0\|_2 = \|u_0\|_2. $$ That is, if the initial data lies in $L^2$, then at each time $t$, the solution (considered as a function in $x$) also lies in $L^2$. The operator $U(t)$ simply pushes the data around the closed ball in $L^2$ of radius $\|u_0\|_2$; the $L^2$ norm of the solution is conserved.

That's nice, but what happens as $t\rightarrow0$ with the solution? Do you actually recover the initial data? One would hope! What do you mean by recover? Ah, a convergence issue!

In the $L^2$-sense, the above calculation will show that $U(t+h)u_0 \rightarrow U(t)u_0$ as $h\rightarrow0$. In particular, this will hold for $t=0$ and we simply note that $U(0)u_0=u_0$. Here are some details: \begin{align*} \|U(t+h)u_0 - U(t)u_0\|_2 &= \|e^{i(t+h)\xi^3}\hat{u}_0 - e^{it\xi^3}\hat{u}_0\|_2 \\ &= \|e^{it\xi^3}[e^{ih\xi^3}-1]\hat{u}_0\|_2 \\ &= \|[e^{ih\xi^3}-1]\hat{u}_0\|_2. \end{align*} Now apply the Lebesgue Dominated Convergence Theorem to the final line to reach the conclusion. This result is abbreviated by saying $$ u(t) \in C(\mathbb{R} : L^2(\mathbb{R})), $$ which means the solution to the IVP describes a continuous curve $t \mapsto u(t)$ in $L^2$, where we have suppressed the spatial variable.

Two comments:

• We can replace $L^2$ with the Sobolev spaces $H^s$ above.
• By as standard theorem in analysis, we can extract a sequence $t_k \rightarrow 0$ so that $U(t_k)u_0 \rightarrow u_0$ point wise a.e. in $x$. But do we generally get pointwise convergence? Here's an answer for the Schödinger equation, where the full story is still actively researched.

One of the main advantages of $L^p$ spaces (for $1<p<\infty$) over the more "traditional" spaces like $C^k([a,b])$ is that the $L^p$ spaces are reflexive, i.e. the double dual $(L^p)''$ is (with an obvious identification) again given by $L^p$.

This implies many useful results, one of which is that for every bounded sequence $(f_n)_n$ in $L^p$, we can extract a subsequence $(f_{n_k})_k$ that converges weakly to some $f \in L^p$. This means that $\varphi(f_{n_k}) \rightarrow \varphi(f)$ for all bounded linear functionals $\varphi$ on $L^p$. By duality theory for $L^p$, these are always given by integration against some function $g_\varphi \in L^{p'}$, where $p'$ is the conjugate exponent determined by $\frac{1}{p} + \frac{1}{p'} = 1$.

This result is useful for many results, for example in the calculus of variations.

Here, one generally considers some (usually nonlinear) functional $\Phi : L^p \rightarrow \Bbb{R}$ and wants to show the existence of a minimizer in some class like $\Gamma := \{f \in L^p \mid \int_0^1 f dx = 0\}$.

[Sidenote: Usually, one considers the Sobolev spaces $W^{k,p}$ instead of the Lebesgue spaces, but these are built on the $L^p$ spaces, so that for example the reflexivity also applies to $W^{k,p}$.

In the class of Sobolev functions one could e.g. take $\Gamma := \{f \in W^{1,p}([0,1]) \mid f(0) = f(1) = 0\}$.]

One then tries to show (using properties of $\Phi$) that for a sequence $(f_n)_n \in \Gamma$ with $\Phi(f_n) \rightarrow \inf_\Gamma \Phi$, the sequence $(f_n)_n$ is already bounded and then extracts $(f_{n_k})_k$ with $f_{n_k} \rightarrow f$ weakly.

One then tries to show (using properties of $\Gamma$ and $\Phi$) that $f \in \Gamma$ as well as $\Phi(f) = \inf_\Gamma \Phi$.

If all this can be done, the existence of a minimizer (namely $f$) follows.

One can then often use this method to obtain solutions to (partial) differential equations, because a minimizer of a functional (which could be given by $\int_0^1 F(x, f(x), f'(x)) dx$ for example) will satisfy the so-called Euler-Lagrange-Equations.