Function Spaces and Foundations

Overview¶

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

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.