Elementary problems solved with Functional Analysis Many times people come and ask me what Functional Analysis is used for and why it's interesting. Of course interest is a matter of taste, and I for one love the subject as it is. There are far reaching applications to Physics, PDE, other areas of analysis and other advanced subjects that I wouldn't be able to demonstrate say to a first or even second year undergraduate student.
What about examples that can be stated in very simple terms, and are somehow "familiar" to the broader audience? I am not aware of classical or definitive examples, so I wanted to ask:
What are some (preferably mathematical) applications of Functional Analysis, that are as elementary as possible? 
 A: My favorite application: Image/video/sound compression and denoising (and more generally, signal processing).
From Fourier to Wavelet decomposition of signals, there is a plethora of techniques stemming from functional analysis for compressing and understanding signals. Examples:


*

*Fourier analysis is the basis for JPEG compression (everyone knows JPEG, right?).

*Wavelet and Fourier are both used in video compression.

*Fourier is used a lot in sound denoising and filtering.

*etc...

A: From W.W. Sawyer's book A Path To Modern Mathematics: We seek  the solution to $f(x)=1+\int_0^x f(t)dt.$ We have $(I-\int)f(x)=1$ so $$f(x)=(I-\int\;)^{-1}(1)=$$ $$=(I+\int +\int \int +\int \int \int +...)(1).$$ Now $I(1)=1$ and $\int(1)=\int_0^x1dt=x$  so $\int \int (1)=\int_0^x(t)dt=x^2/2$ and $\int \int \int (1)=x^3/3!$ and so on . So $f(x)=\sum_{n=0}^{\infty}x^n/n!.$
Sawyer writes that the response is often laughter.
A: A very important application is data compression, the theory of wavelets being particulary interesting here.
A: Here's one of my favorites.
Suppose $f:[0,\infty) \to \Bbb R$ is a function with the property that $f(nx) \stackrel{n\to\infty}{\to} 0$, for every $x>0$. Then prove that $\lim_{a \to \infty} f(a) = 0$.
It seems to be an elementary problem but I don't know how to do it without Baire's Category Theorem. Specifically fix $\epsilon>0$ and let $U_n = \{ x>0 : k \ge n$ implies $|f(kx)|<\epsilon\}$. These are open and since $\bigcup_n U_n = [0,\infty)$, so by Baire some $U_n$ contains an open interval $[b,c]$. Then for $a \in \bigcup_{k\ge n} [kb,kc]$ it holds that $|f(a)|<\epsilon$. But $\bigcup_{k \ge n} [kb,kc]$ contains the open ray $[nb,\infty)\cap [\frac{b}{b-c},\infty)=[\max\{nb,\frac{b}{c-b}\},\infty).$
