The book now pivots from integration as a numeric operation to integration as a foundation for function spaces.

The message: once you have a robust integral, you can treat functions as vectors, define norms, and do analysis with completeness and convergence in a principled way.

8.1 Normed spaces

A normed vector space formalizes the idea of “size” and “distance” between vectors.

In analysis, the key is that convergence in norm implies strong control (much stronger than pointwise convergence).

8.2 Banach spaces

A Banach space is a complete normed space: every Cauchy sequence converges.

Completeness is not a luxury; it’s what guarantees that limits of approximations stay inside the space.

8.3 Completion of spaces

Many naturally occurring spaces are not complete.

Completion is the systematic way to add missing limit points while preserving the metric/norm structure. This is exactly the lens through which L^p spaces are introduced.

8.4 The space L¹

L¹ is the space of (equivalence classes of) integrable functions with norm

  • ‖f‖₁ = ∫ f .

Important idea: functions that differ only on a null set are treated as the same object. This is what makes the norm well-defined and the space well-behaved.

8.5 The Lebesgue spaces L^p

Generalizing L¹:

  • ‖f‖ₚ = (∫ f ^p)^{1/p}, 1 ≤ p < ∞
  • ‖f‖_∞ = ess sup f

The standard inequalities appear here:

  • Hölder inequality
  • Minkowski inequality

These inequalities are the “algebra” that makes L^p usable: they control products, sums, and limits.

8.6 Separable spaces

Separability (existence of a countable dense subset) matters for approximation and for connecting abstract spaces back to computation.

In practice, separability tells you whether you can approximate functions with sequences drawn from a manageable family.

8.7 Complex L^p spaces

Same story, but for complex-valued functions. This matters for Fourier analysis and signal-like applications.

8.8 Hardy spaces H^p

Hardy spaces are alternative function spaces tuned for complex analysis and boundary behavior.

Even if you don’t use them directly, it’s useful to see that “integrability + structure” produces many different spaces specialized for different problems.

8.9 Sobolev spaces W^{k,p}

Sobolev spaces incorporate weak derivatives into the norm.

This is one of the main bridges to PDEs and numerical methods: regularity (smoothness) becomes a measurable property.

Mental model

Chapter 8 turns “∫” into a geometry:

  • a norm defines distance,
  • completeness defines stability under limits,
  • inequalities define algebraic control.

Practice checklist

  • Prove (or at least apply) Hölder and Minkowski fluently.
  • Build intuition for L^p nesting: when does L^q ⊂ L^p hold?
  • Get comfortable with “equal a.e.” as an equivalence relation.