TITLE: What is $L^p$-convergence useful for?
QUESTION [6 upvotes]: Why do people care about $L^p$-convergence $f_n \rightarrow f$? Are there any interesting application of $L^p$-convergence? 
For example, if $p=\infty$, then the limit $f$ of the sequence $f_n$ of continuous function is again continuous. are there any other interesting, useful applications?

REPLY [3 votes]: 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.