Mathematics · Frontier · 35 min · Chapter 19 of 28

The frontier: analysis, geometry, and categories

Three doors past the undergraduate core — functional analysis, algebraic geometry, and category theory — stated honestly as definitions and one theorem each.

Read first: Real analysis, Abstract algebra, Topology, Linear algebra

The idea

Finite-dimensional linear algebra ends too soon for analysis. Functions themselves form vector spaces: the set of continuous functions on [0, 1], the set of square-integrable functions, the set of bounded sequences. A norm is a length ||x|| satisfying the usual rules, and a space is Banach when every Cauchy sequence converges inside it. Completeness is what lets you take limits of approximations and stay in the space. A Hilbert space adds an inner product whose norm is ||x|| = √⟨x, x⟩. Orthonormal bases, orthogonal projection, and the spectral theorem for compact self-adjoint operators all survive, with hypotheses you must now state. The spectral theorem is why quantum mechanics can talk about observables as operators and measurement values as eigenvalues, including continuous spectra.

Algebraic geometry begins with a polynomial, or a list of them. The variety V(f₁, …, fₖ) is the set of points where they all vanish. Over the reals, x² + y² + 1 = 0 defines the empty set; over the complexes it defines a curve. Hilbert’s nullstellensatz says that over an algebraically closed field, the ideal of polynomials vanishing on V(I) is the radical of I. Geometry and ideal theory become the same subject. Modern algebraic geometry replaces the variety by a scheme, so the equations can remember multiplicity and can live in families. You do not need schemes to use the idea that a shape defined by equations has an algebraic description of its functions.

Category theory is the language that notices the same diagram in all of the above. A category has objects, morphisms between them, an identity morphism on each object, and an associative composition. Sets and functions form a category. Groups and homomorphisms form a category. Topological spaces and continuous maps form a category. A functor sends objects to objects and morphisms to morphisms, preserving identities and composition. The fundamental group is a functor from pointed spaces to groups. Yoneda’s lemma says an object is determined, up to isomorphism, by the morphisms into or out of it: to know the object, know how the rest of the category sees it. That sentence is the whole spirit of the subject, and it is a precise theorem, not a slogan, once the functor category is defined.

Hold onto this

Functional analysis is linear algebra in infinite dimensions, with limits.
A Banach space is a complete normed vector space. A Hilbert space also has an inner product.
Algebraic geometry studies shapes cut out by polynomial equations, and it lets those equations live over any field.
A category is objects and arrows with composition. Functors are the maps between categories.

Formulas

Norm

∥ x ∥ \geq 0, ∥ c x ∥ = ∣ c ∣ ∥ x ∥, ∥ x + y ∥ \leq ∥ x ∥ + ∥ y ∥

Length: positive, scales, and satisfies the triangle inequality. Zero only at the zero vector.

||x||: The norm of a vector

Inner product

⟨ a x + b y, z ⟩ = a ⟨ x, z ⟩ + b ⟨ y, z ⟩, ⟨ x, x ⟩ \geq 0

A generalization of the dot product. It defines a norm by ||x|| = √⟨x, x⟩.

⟨ , ⟩: Inner product, linear in the first slot in the convention used here

Affine variety

V (f) = {(x_{1}, \dots, x_{n}) : f (x_{1}, \dots, x_{n}) = 0}

The shape cut out by one polynomial equation. Several polynomials give an intersection of such shapes.

f: A polynomial

Functor

F (id_{A}) = id_{F (A)}, F (g \circ f) = F (g) \circ F (f)

A functor preserves identities and composition. It is a homomorphism of categories.

F: A (covariant) functor

Banach fixed-point theorem

d (T x, T y) \leq k d (x, y), 0 \leq k < 1

A contraction on a nonempty complete metric space has a unique fixed point, and iteration converges to it from anywhere.

T: The map
k: A uniform contraction constant strictly less than 1

Worked examples

1. A contraction finds a root

On the reals, consider T(x) = cos x. Why does iteration converge?

By the mean value theorem, |cos x − cos y| = |sin c| |x − y| ≤ k |x − y| is not quite global, because |sin c| can be 1.
On the interval [0, 1], the derivative of cos is −sin, and |sin x| ≤ sin 1 < 0.85, so T maps a neighborhood of its fixed point to itself and is a contraction there.
Starting at 0 and iterating cosine converges to the unique solution of cos x = x, the Dottie number, about 0.739.

Iteration converges to the unique fixed point of cosine near 0.739. The theorem supplies existence, uniqueness, and the algorithm.

2. A variety you already know

What is V(x² + y² − 1) over the reals?

The equation x² + y² = 1 is the unit circle.
Over the complexes the same equation defines a complex curve, a different and larger object.

The unit circle in the plane. Algebraic geometry’s first examples are the curves of high-school analytic geometry.

Try these

Why is the set of polynomials not a Banach space under the usual “size of coefficients” attempts people first try, and what is the real obstruction people mean?

Name three categories you have already used in this book.

Common mistakes

Infinite-dimensional spaces do not have the theorem “a linear injection from V to V is a surjection.” That equivalence is finite-dimensional.
The real and complex points of the same equations can be wildly different. Always say which field you are working over.
Category theory does not replace the earlier subjects. It organizes them. A functor you cannot compute on a single example is a definition you have not understood.