Why is it important that $L^P$ spaces be complete? I know that Banach spaces are ubiquitous and incredibly important in a lot of areas of math, but I was hoping for an intuitive explanation as to why (and when) it's important in the case of $L^p$ spaces.
My understanding is that if we're interested in any limiting process in an $L^p$ space, completeness ensures that the limits that should be there are actually in the space. Is there anything more to it? Any good examples or well-known theorems that rely on the completeness of $L^p$?
 A: It is very difficult to do analysis on a space if it is not complete. Consider for instance $\mathbb{Q}$. The function $f\colon[0,2]\cap\mathbb{Q}\to\mathbb{Q}$ defined as
$$
f(x)=\frac{1}{x^2-2}
$$
is continuous, but is not bounded.
An important result in Banach spaces (and in particular in $L^p$) that depends on completeness is:
Theorem. If $X$ is a Banach space with norm $\|\ \|$, $\{x_n\}\subset X$, and $\sum\|x_n\|<\infty$, then $\sum x_n$ converges.
A: 
Why is it important that $L^p$ spaces be complete?

Among other things, because we can use it to prove existence of solutions for some problems. For instance, we use (among other things) the completeness of $L^p$ to define a certain Banach space $X$ and to prove that the operator
\begin{align*}
A&=\begin{pmatrix}
0 & 1 & 0 & 0 & 0 & 0 & 0\\ 
\frac{k}{\rho_1}\partial^2_x-\frac{k_0l^2}{\rho_1}I & 0 & \frac{k}{\rho_1}\partial_x & 0 & \frac{l(k+k_0)}{\rho_1}\partial_x & 0 & 0\\ 
0 & 0 & 0 & 1 & 0 & 0 & 0\\
-\frac{k}{\rho_2}\partial_x & 0 & \frac{b}{\rho_2}\partial^2_x-\frac{k}{\rho_2}I & 0 & -\frac{kl}{\rho_2}I & 0 & -\frac{\gamma}{\rho_2}\partial_x\\  
0 & 0 & 0 & 0 & 0 & 1 & 0\\ 
-\frac{l(k_0+k)}{\rho_1}\partial_x & 0 & -\frac{lk}{\rho_1}I & 0 & \frac{k_0}{\rho_1}\partial^2_x-\frac{l^2k}{\rho_1}I & 0 & 0\\ 
0 & 0 & 0 & -m\partial_x & 0 & 0 & k_1\partial_x^2
\end{pmatrix}
\end{align*}
(defined in a suitable subspace of $X$) is the infinitesimal generator of a $C_0$-semigrouop of contractions on $X$. As a consequence, we get existence and uniqueness of solution for the problem
$$\begin{align}
\rho_{1}\varphi_{tt} - k(\varphi_{x}+\psi+lw)_{x} - k_{0}l(w_{x}-l\varphi)&=0\\
\rho_{2}\psi_{tt} - b\psi_{xx}+k(\varphi_{x} + \psi+lw)+\gamma\theta_x&=0\\
\rho_{1}w_{tt} - k_{0}(w_{x}-l\varphi)_{x} + kl(\varphi_{x}+\psi+lw)&=0\\
\theta_t-k_1\theta_{xx}+m\psi_{tx}&=0
\end{align}\tag{1}$$
with appropriated boundary and initial conditions. In addition, we also use (among other things, again) the completeness of $L^p$ to prove that the solution decays exponentially. This is an interisting result because system $(1)$ come from the study of thermoelastic curved beams.
