Chapter 1
Hilbert Spaces
The geometric setting for everything that follows: a vector space with an inner product, complete with respect to the norm that inner product induces.
Inner product spaces
Definition
Let $H$ be a vector space over $\mathbb{C}$. An inner product is a map
$\langle \cdot, \cdot \rangle : H \times H \to \mathbb{C}$ satisfying, for all $x, y, z \in H$
and $\alpha \in \mathbb{C}$:
- $\langle x, x \rangle \geq 0$, with equality iff $x = 0$ (positive-definiteness);
- $\langle x, y \rangle = \overline{\langle y, x \rangle}$ (conjugate symmetry);
- $\langle \alpha x + y, z \rangle = \alpha \langle x, z \rangle + \langle y, z \rangle$ (linearity in the first argument).
Every inner product induces a norm via $\|x\| = \sqrt{\langle x, x \rangle}$. The two most important inequalities relating the inner product and this norm are:
Cauchy–Schwarz inequality
$$ |\langle x, y \rangle| \leq \|x\| \, \|y\| \qquad \text{for all } x, y \in H, $$
with equality if and only if $x$ and $y$ are linearly dependent.
Parallelogram law
$$ \|x + y\|^2 + \|x - y\|^2 = 2\|x\|^2 + 2\|y\|^2. $$
A normed space arises from an inner product exactly when its norm satisfies this identity
(Jordan–von Neumann theorem).
Completeness and the definition of a Hilbert space
Definition
A Hilbert space is an inner product space that is complete with
respect to the induced norm — that is, every Cauchy sequence in $H$ converges to a limit
in $H$.
Standard examples used throughout this site:
- $\mathbb{C}^n$ with $\langle x, y \rangle = \sum_{i=1}^n x_i \overline{y_i}$ — finite-dimensional, always complete.
- $\ell^2 = \left\{ (a_n)_{n\geq 1} : \sum_n |a_n|^2 < \infty \right\}$, with $\langle a, b \rangle = \sum_n a_n \overline{b_n}$ — the prototypical infinite-dimensional separable Hilbert space.
- $L^2([a,b]) = \left\{ f : \int_a^b |f(x)|^2 \, dx < \infty \right\}$, with $\langle f, g \rangle = \int_a^b f(x)\overline{g(x)} \, dx$.
Orthogonality and orthonormal bases
Vectors $x, y \in H$ are orthogonal, written $x \perp y$, if $\langle x, y \rangle = 0$. A subset $\{e_i\}_{i \in I}$ is orthonormal if $\langle e_i, e_j \rangle = \delta_{ij}$, and an orthonormal basis if in addition its closed linear span is all of $H$.
Parseval's identity
If $\{e_i\}_{i \in I}$ is an orthonormal basis of $H$, then every $x \in H$ satisfies
$$ x = \sum_{i \in I} \langle x, e_i \rangle e_i, \qquad
\|x\|^2 = \sum_{i \in I} |\langle x, e_i \rangle|^2, $$
where the sum has at most countably many nonzero terms.
Why this matters for operators
Orthonormal bases let us represent a bounded operator on a separable Hilbert space by an
(infinite) matrix $T_{ij} = \langle Te_j, e_i \rangle$, and they are the key tool behind the
spectral theorem for compact self-adjoint operators (see
Chapter 7).