Linear algebra studies finite lists of numbers: vectors with two, three, a hundred components. Functional analysis asks the daring next question — what if a vector had infinitely many components? What if a whole function were a single point in space? This is the mathematics of infinite-dimensional spaces, where the objects being added and scaled are functions, sequences and signals.

It is where linear algebra, calculus and analysis fuse into one subject. The payoff is enormous: functional analysis is the native language of quantum mechanics, of differential equations, of Fourier analysis and signal processing, and of the modern theory of optimisation. When you want to treat "the space of all solutions" as a geometric object, this is the toolkit.

The big idea: geometry in infinitely many dimensions

One thread runs through everything here. Take the familiar ideas of length, distance, angle and completeness — and carry them faithfully into spaces where a "point" is an entire function. Most of the difficulty (and all of the beauty) comes from a single new hazard: infinite dimensions leak. Sequences that should converge can escape the space; the unit ball stops being compact. Taming that leakage — with norms, completeness and clever theorems — is the whole game.

The shape of the journey

This course moves in three stages, each building on the last.

• Stage A — Spaces. Giving vectors a length, then demanding that convergent sequences have somewhere to land: normed and complete (Banach) spaces.
• Stage B — Operators and duals. Linear maps between these spaces, the space of measurements (the dual), and the great structural theorems that make them work.
• Stage C — Hilbert spaces and spectra. The best-behaved spaces, where angles return — and the spectral theory that generalises eigenvalues to infinite dimensions.

Stage A — Spaces

1. Normed Vector Spaces
2. Banach Spaces

Stage B — Operators and duals

1. Bounded Linear Operators
2. The Dual Space
3. The Hahn–Banach Theorem
4. The Baire Category Theorem

Stage C — Hilbert spaces and spectra

1. Hilbert Spaces Revisited
2. Compact Operators
3. Spectral Theory of Operators

Let's get started

We begin by giving an abstract vector a length. That single number — the norm — unlocks distance, convergence and every piece of geometry that follows.

Let's get started → Normed Vector Spaces