Hilbert space · aiwiki.page

A Hilbert space is a vector space equipped with an inner product and complete with respect to the distance that this inner product defines. It combines geometric ideas—length, perpendicularity, and projection—with the convergence properties needed to handle infinite sequences and series. Hilbert spaces may be finite-dimensional or infinite-dimensional; their elements can be coordinate vectors, sequences, or functions. They provide a framework connecting linear algebra with infinite-dimensional analysis. (ocw.mit.edu)

Definition and completeness

A Hilbert space $H$ is defined over the real numbers or complex numbers. Its inner product $\langle x,y\rangle$ is positive definite and conjugate symmetric, and is linear in one argument and conjugate-linear in the other. Here the convention is linearity in the first argument; many physics texts use the opposite convention. The induced norm and distance are

\|x\|=\sqrt{\langle x,x\rangle}, \qquad d(x,y)=\|x-y\|.

Completeness means that every Cauchy sequence in this distance converges to an element of $H$ . Thus, whenever vectors become arbitrarily close to one another, their limiting vector remains within the space. An inner product space without the completeness requirement is often called a pre-Hilbert space. (ocw.mit.edu)

Every Hilbert space is a normed vector space and a complete metric space. A complete normed vector space is called a Banach space, but not every Banach-space norm comes from an inner product. Inner-product norms satisfy the parallelogram identity,

\|x+y\|^2+\|x-y\|^2=2\|x\|^2+2\|y\|^2.

Conversely, this identity characterizes norms induced by inner products. (ocw.mit.edu)

Principal examples

Finite-dimensional examples include $\mathbb R^n$ and $\mathbb C^n$ , with

\langle x,y\rangle=\sum_{j=1}^{n}x_j\overline{y_j}.

These spaces are complete. The sequence space $\ell^2$ consists of all sequences $x=(x_j)_{j\geq1}$ satisfying

\sum_{j=1}^{\infty}|x_j|^2<\infty,

with inner product defined by the corresponding infinite sum. It is a basic infinite-dimensional example. (ocw.mit.edu)

In measure theory, the space $L^2(X,\mu)$ consists of equivalence classes of measurable functions whose squared absolute values have finite integral. Its inner product is

\langle f,g\rangle=\int_X f\,\overline g\,d\mu.

Functions equal almost everywhere represent the same element: this identification ensures that zero norm implies the zero element. Completeness distinguishes $L^2$ from many smaller inner product spaces. For example, continuous functions on $[0,1]$ , equipped with the $L^2$ norm, do not form a complete space. (ocw.mit.edu)

Orthogonality and bases

Vectors $x$ and $y$ are orthogonal when $\langle x,y\rangle=0$ . The Pythagorean theorem then takes the form $\|x+y\|^2=\|x\|^2+\|y\|^2$ . The Cauchy–Schwarz inequality,

|\langle x,y\rangle|\leq\|x\|\,\|y\|,

controls inner products and underlies many convergence arguments. (ocw.mit.edu)

An orthonormal basis is an orthonormal family whose linear span is dense in the space. In a separable Hilbert space—one possessing a countable dense subset—such a basis can be chosen finite or countably infinite. For an infinite basis $(e_n)$ , every vector has the expansion

x=\sum_{n=1}^{\infty}\langle x,e_n\rangle e_n, \qquad \|x\|^2=\sum_{n=1}^{\infty}|\langle x,e_n\rangle|^2.

The series converges in norm, and the second equation is Parseval’s identity. Unlike an algebraic basis, an orthonormal basis generally requires infinite rather than finite expansions. These identities underpin the Hilbert-space treatment of Fourier series. Every infinite-dimensional separable Hilbert space is isometrically isomorphic to $\ell^2$ over the same scalar field. (ocw.mit.edu)

Projection and representation

For every closed linear subspace $M\subseteq H$ , each vector $x$ has a unique decomposition

x=P_Mx+z,\qquad P_Mx\in M,\quad z\in M^\perp.

The vector $P_Mx$ uniquely minimizes $\|x-m\|$ over $m\in M$ . This orthogonal projection theorem extends the geometry of dropping a perpendicular to infinite dimensions. Closedness matters: a nonclosed subspace need not contain a nearest point. (ocw.mit.edu)

The Riesz representation theorem states that every bounded linear functional $F:H\to\mathbb F$ has a unique representing vector $y$ such that

F(x)=\langle x,y\rangle, \qquad \|F\|=\|y\|.

It identifies the continuous dual space with the Hilbert space itself, conjugate-linearly under the convention used here in the complex case. Both results depend crucially on completeness. (ocw.mit.edu)

Operators

A bounded linear operator $T:H\to H$ has a unique adjoint $T^*$ , characterized by

\langle Tx,y\rangle=\langle x,T^*y\rangle.

In finite dimensions, its matrix is the conjugate transpose of the matrix of $T$ . Self-adjoint operators satisfy $T=T^*$ . Hilbert-space operator theory extends matrix methods to spaces where vectors may be functions and operators may involve integration or differentiation. Unbounded operators require explicit attention to their domains. (ocw.mit.edu)

Applications

In quantum mechanics, pure states correspond to rays in a complex Hilbert space: normalized vectors differing only by an overall phase represent the same state. A particle’s wave function commonly belongs to an $L^2$ space. Observables are represented by self-adjoint operators, and composite systems use Hilbert-space tensor products. The inner product supplies transition amplitudes, whose squared magnitudes determine probabilities through the Born rule. (damtp.cam.ac.uk)

In machine learning, a reproducing kernel Hilbert space is a Hilbert space of functions in which point evaluation is continuous. Evaluation can therefore be represented by an inner product with a kernel function. This structure supports kernel methods, allowing computations with potentially infinite-dimensional feature representations through kernel values rather than explicit coordinates. Not every Hilbert space of functions has this property. (stat.berkeley.edu)