Hilbert Space

Introduction

A Hilbert space is an abstract vector space possessing the structure of an inner product that allows length and angle to be measured. Furthermore, Hilbert spaces are complete: there are enough limits in the space to allow the techniques of calculus to be used.

Definition

Formally, a Hilbert space is a real or complex inner product space that is also a complete metric space with respect to the distance function induced by the inner product. To say that an inner product space is a Hilbert space means that it is complete, or closed, in the sense that if a sequence of vectors in the space converges to a limit in the metric induced by the inner product, then that limit is also in the space.

A visual representation of a Hilbert space, showing vectors and their inner products.

Properties

Hilbert spaces are of key importance in the mathematical formulation of quantum mechanics, where states of a quantum system are represented by vectors in a Hilbert space. They also play a role in the study of partial differential equations and in mathematical analysis.

Completeness

The defining characteristic of a Hilbert space is its completeness. This means that a Cauchy sequence of vectors in the space will always converge to a limit that is also in the space.

Orthogonality

In a Hilbert space, the concept of orthogonality is defined using the inner product. Two vectors are orthogonal if their inner product is zero. This concept is a generalization of the concept of perpendicularity in Euclidean space.

Basis and Dimension

Every Hilbert space has an orthonormal basis, and the cardinality of this basis is called the dimension of the space. The dimension can be finite or infinite.

Applications

Hilbert spaces have many applications in mathematics and physics. They are used in quantum mechanics, where the state of a quantum system is represented by a vector in a Hilbert space. They are also used in the study of partial differential equations, in signal processing, and in statistics.

Quantum Mechanics

In quantum mechanics, the state of a quantum system is represented by a vector in a Hilbert space. The inner product of two vectors represents the probability amplitude for transitioning between the two states represented by the vectors.

Signal Processing

In signal processing, Hilbert spaces are used to represent signals and to define the Hilbert transform, which is a linear operator that shifts the phase of a signal.

Statistics

In statistics, Hilbert spaces are used in the theory of statistical inference, particularly in the study of hypothesis testing and estimation.