Why is it important that $L^P$ spaces be complete?

Question

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 $L^p$?

Answer 1

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.

Answer 2

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.