Hilbert Spaces

We now have a complex vector space with an inner product. A Hilbert space is such a space with one extra technical guarantee — completeness — that makes infinite-dimensional quantum mechanics well-behaved. This is the precise mathematical setting in which all of quantum theory is formulated.

From inner-product space to Hilbert space

A Hilbert space $\mathcal{H}$ is a complex inner-product space that is complete with respect to the norm induced by the inner product. The vectors are still kets $|\psi\rangle$ , the inner product $\langle \phi | \psi \rangle$ still measures overlap, and the norm is still $\|\psi\| = \sqrt{\langle \psi | \psi \rangle}$ . The new word is "complete."

What completeness means

A sequence of vectors $\{|\psi_n\rangle\}$ is a Cauchy sequence if its terms eventually get arbitrarily close together: for every $\varepsilon > 0$ there is an $N$ such that

\| \psi_m - \psi_n \| < \varepsilon \quad \text{whenever } m, n > N.

The space is complete if every Cauchy sequence actually converges to a limit that lives inside the space:

\lim_{n \to \infty} \|\psi_n - \psi\| = 0 \quad \text{for some } |\psi\rangle \in \mathcal{H}.

In words: there are no "holes." If a sequence of states is settling down, the thing it settles down to is itself a legitimate state. This is the same property that distinguishes the real numbers $\mathbb{R}$ (complete) from the rationals $\mathbb{Q}$ (incomplete — a sequence of rationals can converge to an irrational like $\sqrt{2}$ , which is not rational).

Finite dimensions: completeness is free

Every finite-dimensional complex inner-product space is automatically a Hilbert space, because $\mathbb{C}^n$ inherits completeness from $\mathbb{C}$ . This is why the Hilbert-space machinery rarely needs to be invoked explicitly when working with qubits: a single qubit lives in $\mathbb{C}^2$ , and $n$ qubits in $\mathbb{C}^{2^n}$ , both of which are complete with no extra effort. The full apparatus of completeness becomes essential only for continuous systems.

Infinite dimensions: where completeness earns its keep

A particle on a line has a wavefunction $\psi(x)$ , and its state space is $L^2(\mathbb{R})$ — the space of square-integrable functions, those with

\langle \psi | \psi \rangle = \int_{-\infty}^{\infty} |\psi(x)|^2 \, dx < \infty.

The inner product of two such functions is

\langle \phi | \psi \rangle = \int_{-\infty}^{\infty} \overline{\phi(x)}\, \psi(x) \, dx.

$L^2(\mathbb{R})$ is complete, which is precisely what lets us expand any wavefunction in an infinite basis (energy eigenstates, Fourier modes) and trust that the infinite sum converges back to a genuine state. Without completeness, limits of physical states could fail to be states, and the spectral theorem — the foundation for measuring observables — would not hold.

Separability and a countable basis

The Hilbert spaces of physics are separable: they admit a countable orthonormal basis $\{|e_1\rangle, |e_2\rangle, \ldots\}$ . Any state can then be written as a (possibly infinite) sum

|\psi\rangle = \sum_n c_n |e_n\rangle, \qquad c_n = \langle e_n | \psi \rangle,

and completeness guarantees this series converges in norm to $|\psi\rangle$ whenever $\sum_n |c_n|^2 < \infty$ . This expansion — and the convergence guarantee behind it — is what we will use throughout the rest of the module.

The takeaway

A Hilbert space is a complete complex inner-product space. Completeness ensures convergent sequences of states have limits that are still states, which is automatic in finite dimensions and indispensable in infinite ones. Every quantum system, from a qubit to a free particle, has a Hilbert space as its state space.