TITLE: Elementary problems solved with Functional Analysis
QUESTION [7 upvotes]: 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?

REPLY [3 votes]: 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.