Function Spaces and Foundations - Applied Functional Analysis

Big Idea

PDEs look like ODEs—if you squint. The key insight of functional analysis is to treat solutions as points in infinite-dimensional function spaces, where differential operators become linear maps between Banach spaces. This abstraction lets us port finite-dimensional intuition (eigenvalues, convergence, compactness) to the PDE world.

Overview¶

Why do we need functional analysis? Consider a reaction-diffusion equation:

\frac{\partial u}{\partial t} = d \Delta u + f(u)

(1)

If we set $A[u] := d\Delta u$ , this looks like the ODE $\dot{x} = Ax + f(x)$ . For ODEs, solutions live in $\mathbb{R}^n$ —a finite-dimensional space with a well-understood theory. But PDE solutions live in infinite-dimensional function spaces, and much of finite-dimensional intuition breaks down:

Not all norms are equivalent.
Bounded closed sets need not be compact.
Convergence comes in many flavors (strong, weak, weak-*).

This chapter sets up the language and tools needed to work in these spaces: norms, completeness, density, separability, and compactness.

What You Will Learn¶

How to define and compare norms on function spaces ( $L^p$ , $C^k$ , Sobolev).
Completeness and Banach spaces—why we need them and how to verify them.
Density and approximation—mollifiers, Weierstrass approximation.
Compactness in infinite dimensions—Arzelà–Ascoli and why bounded sequences don’t always have convergent subsequences.