Definition

Noun: A Hilbert space is a specific, complete metric space that possesses the structure of an inner product space. This inner product allows for the definition of angles and lengths (norms), making it a generalization of Euclidean space to potentially infinite dimensions. Its completeness means that every Cauchy sequence of vectors in the space converges to a limit that is also within the space.

Examples

• The set of all square-integrable functions forms a classic example of an infinite-dimensional Hilbert space.
• In quantum mechanics, the state of a system is represented by a vector in a Hilbert space.
• Finite-dimensional Euclidean space, (\mathbb{R}^n) with the standard dot product, is the simplest example of a Hilbert space.

Advanced Usage

• Separable Hilbert Space: A Hilbert space that contains a countable, dense subset. Most Hilbert spaces used in applications (like quantum mechanics and signal processing) are separable.
• Reproducing Kernel Hilbert Space (RKHS): A Hilbert space of functions where point evaluation is a continuous linear functional. This property is crucial in machine learning and functional analysis.

Variants and Related Words

• Inner Product Space: A vector space equipped with an inner product. A Hilbert space is a complete inner product space.
• Banach Space: A complete normed vector space. Every Hilbert space is a Banach space, but the converse is not true, as a Banach space's norm does not necessarily come from an inner product.
• Metric Space: A set where a distance (metric) is defined between any two points. A Hilbert space is a specific type of metric space where the metric is induced by the norm from the inner product.

Synonyms

• Complete inner product space

Related Phrases and Concepts

• Orthonormal basis: A set of mutually orthogonal unit vectors that span a dense subspace of the Hilbert space.
• Projection theorem: In a Hilbert space, for any closed convex set and any point, there exists a unique point in the set closest to it.
• Riesz representation theorem: A fundamental result stating that every continuous linear functional on a Hilbert space can be uniquely represented as an inner product with a fixed vector in that space.

Noun

1. a metric space that is linear and complete and (usually) infinite-dimensional