TITLE: What is functional analysis in simple words?
QUESTION [38 upvotes]: To begin with , I am only a secondary school student (17yo) but I am very interested in higher mathematics. However we only learn so little in my school (only single variable calculus and basic linear algebra). In the past I have self-learnt some abstract algebra and very basic topology by finding online resources, but I can never get deep into those subjects.
When I read about functional analysis, I encounter objects like function spaces and infinite-dimensional spaces which I can never understand. What does it exactly mean to be a function space, how do you measure metric? I know it is hard and requires much real analysis. Can anyone give me some easy ideas and introductions?

REPLY [3 votes]: Algebraic analysis is finding an unknown [ function ] in terms of an infinite polynomial.  The unknown function is specified by some kind of differential equation.
Functional analysis is finding an unknown in terms of an infinite series of functions.  The simplest example being 'fourier analysis' which is a general solution of the empty-space wave equation.
Why would we do this ? Surely it involves much more computation, especially if the end result has to be evaluated numerically; as needed in experimental physics and engineering problems ?  Remember, an infinite series of transcendental functions, themselves with infinite representations, slow to evaluate numerically.
In some cases, the basis functions may be easy - such as Legendre Polynomials, which often appear in quantum theory; other times difficult  functions such as Bessel Functions, which aren't well understood by even many undergraduate level mathematicians.
Even the simplest case, fourier series, in the early days pure mathematicians were unsure if a fourier decomosition of an unknown function was an accurate representation of it, and under what conditions - what ranges of dependent variables were safe ? And how many terms to use for a specified degree of accuracy ?
Later mathematicians Sturm and Liouville showed that for ALL second order linear differential equations , which are the vast majority used in Science and Engineering, the basis functions also known as eigenfunctions are are always orthogonal, linear and the functional decomposition of any solution is an unique and accurate representation of the true solution.
Further, eigenfunctions always have recurrence relations a good example being the Tchebyshef polynomials which help in further analysis, both algebraic and numerical.