If L^p spaces are the general normed/Banach story, L² is the special case where geometry becomes inner-product geometry. That extra structure unlocks orthogonality, projections, and Fourier methods.

9.1 Hilbert spaces

A Hilbert space is a complete inner product space.

The inner product ⟨f, g⟩ induces the norm:

  • ‖f‖ = √⟨f, f⟩

In L², the inner product is typically:

  • ⟨f, g⟩ = ∫ f(x) \overline{g(x)} \, dμ(x)

Completeness ensures that orthogonal expansions converge to a limit inside the space.

9.2 Orthogonal sets

Orthogonality is the engine for “coordinate systems” in infinite dimensions.

Key working tools:

  • orthonormal sets
  • Fourier coefficients (projections)
  • Pythagorean identity and Bessel-type inequalities

The big idea: with an orthonormal basis, approximation becomes projection onto finite-dimensional subspaces.

9.3 Classical Fourier series

Fourier series are presented as the archetype:

  • represent a function (in L² sense) as a sum of sines/cosines.

Important nuance:

  • convergence is typically in the L² norm (mean-square), not necessarily pointwise.

This is a recurring theme in applied math: “energy convergence” is often the right notion.

9.4 The Sturm–Liouville problem

Sturm–Liouville theory produces orthogonal families from differential equations.

Why it belongs here:

  • it explains where many classical orthogonal bases come from (eigenfunction expansions)
  • it connects functional analysis back to PDE/physics

9.5 Other bases for L²

The chapter closes by showing that trigonometric functions are not the only game in town.

Depending on boundary conditions and domains, you may prefer:

  • orthogonal polynomials
  • eigenfunction systems
  • or other structured bases

Mental model

Hilbert space theory is what makes “least squares,” “best approximation,” and “spectral methods” rigorous.

Once you see L² as a geometric space, many results become intuitive:

  • approximation = projection
  • error = orthogonal residual
  • expansions = coordinates

Practice checklist

  • Compute Fourier coefficients as inner products.
  • Practice proving orthogonality and using it to simplify integrals.
  • Interpret convergence in L² vs pointwise convergence.