Introduction

Grading Policy

We will have 2 mid-semester examinations worth 20 marks each. There will be a quiz during most tutorial sessions worth 4 marks; this quiz will be based on assignment questions given during that week. The best five quizzes will be counted towards another 20 marks. The final end-semester examination will be worth 40 marks out of which 10 marks will be entirely based on the assignments.

In each assignment, there will be some challenging (starred) problems. Solving these problems should be its own reward! However, if you do solve these problems you will be better prepared to do the challenging portion of the end-semester examination (which will be worth about 20-30 marks). Elegant and/or original solutions to the challenging problems will get a special mention.

What is functional analysis?

A functional is a function on a space of functions, and so functional analysis is the analysis of such functions. The study of functions on a space is closely tied to the study of the space itself, hence we can also think of functional analysis as the study of spaces of functions, also called "function spaces".

Why do we want to study spaces of functions? One motivation can come from trying to understand why we study real numbers.

We have equations like x² = 2 and cos(x)=0 which do not have solutions in rational numbers. So we bring in real numbers as a way of finding "approximate" solutions. Note that an approxiate solution is not a single solution, but a sequence of better and better solutions (or at least an algorithm to construct such a sequence). For example, we start with p₀/q₀ = 1/1 and iteratively define
$p_{n+1}/q_{n+1} = \frac{p_n^2 + 2q_n^2}{2p_nq_n}$
This gives a sequence of rational numbers that form a better and better approximation to a root of x² = 2.

Note that, even to understand this statement, we need to understand what we mean by better and better! We know that this is given by a notion of convergence or metric and the above sequence is a Cauchy sequence in this metric. Moreover, we also need to see that (equivalence classes of) Cauchy sequences have properties similar to those of numbers in order for this approach to extend our number system.

An example

Let us say that we want to solve the functional equation (df/dx)(x)=f(x). To start with the only functions we know are the simplest kind --- polynomials. So we consider the (linear) map D : P ↦ dP/dx from polynomials to polynomials. In these terms, if V denotes that space of polynomials, then D : V → V is a linear transformation and we are looking for an eigenvector with eigen value 1.

Given any polynomial P, repeated differentiation produces 0. Hence, D is a locally nilpotent transformation on V (in other words for every polynomial P, there is a k so that D^k(P)=0). From this it is obvious that there is no such eigenvector in V.

We can define the polynomials $P_n(x)=\sum_{k=0}^n x^n/n!$ and note that D(P_n + 1)=P_n. So, if the following things hold:

1. there is a notion of convergence so that P_n converges (in other words a topology or metric on V in which P_n is a Cauchy sequence).
2. the operator D needs to be continuous with respect to this notion of convergence.

Suppose that (1) and (2) hold. Then, let P_n converge to f then D(P_n)=P_n − 1 also converges to f. On the other hand, by continuity of D, the sequence D(P_n) converges to D(f). Thus we see that D(f)=f. Note that we still need to know that f is a "function". For this, we note that for every a ∈ ℂ, we have the evaluation map e_a : V → ℂ defined by P ↦ P(a). This too should extend to f in order to make sense of f(a). In other words:

3. The map e_a should be continuous.

Thus, we see that in order to solve differential equations, we need to combine linear algebra with continuity. This leads us to the study of topological vector spaces. Furthermore, we need to study such spaces which are spaces of functions.