Hilbert Spaces
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A Hilbert space is a complete inner product space over the real or complex numbers, with the property that it is complete under the norm induced by an inner product. It is named after David Hilbert, who first defined it in his 1896 paper “Über die Inhomogene mit quadratischer Form” (On Non-Orthogonal Spaces of Quadratic Form).
Definition
A Hilbert space \(H\) is a vector space equipped with an inner product \(\langle \cdot, \cdot \rangle\), where the norm is given by \(\| x \| = \sqrt{\langle x, x \rangle}\). The space \(H\) is said to be complete if every Cauchy sequence in \(H\) converges to an element in \(H\).
Properties
Completeness
A Hilbert space is complete under the norm induced by the inner product if and only if it has no infinite dimension. If a Hilbert space is finite-dimensional, then it is automatically complete because every Cauchy sequence converges to a finite linear combination of vectors in the basis.
Equivalences
Two Hilbert spaces \(H\) and \(K\) are said to be equivalent if there exists an invertible linear transformation \(T: H \to K\). If two Hilbert spaces have the same dimension, then they can always be made equivalent by stretching or shrinking one of them appropriately.
Examples
\(\mathbb{R}^n\)
The space \(\mathbb{R}^n\) with the standard inner product is a complete Hilbert space. It has no infinite dimension and can be equipped with an inner product that induces the Euclidean norm.
\(\ell_1\)
The space \(\ell_1\) of absolutely summable sequences of real numbers is a Hilbert space with respect to the inner product given by \(\langle (x_1, x_2), (y_1, y_2) \rangle = \sum_{i=1}^\infty x_i \overline{y}_i\). This space has no infinite dimension and can be equipped with an inner product that induces the \(l_1\) norm.
Notation
- \(H\): Hilbert space
- \(\| x \|\): norm in \(H\)
- \(\langle \cdot, \cdot \rangle\): inner product in \(H\)
- \(\mathcal{H}\): set of all Hilbert spaces
- \(\mathcal{O}(n)\): set of all finite-dimensional linear operators on \(\mathbb{R}^n\)
Applications
Quantum Mechanics
The Hilbert space of a quantum state is used to describe the properties of particles in quantum mechanics. The inner product given by \(\langle \psi | \phi \rangle\) represents the expectation value of an operator acting on the state.
Machine Learning
Hilbert spaces are often used as features spaces in machine learning algorithms, such as Support Vector Machines and Principal Component Analysis (PCA).
History
The concept of Hilbert spaces was first introduced by David Hilbert in his 1896 paper “Über die Inhomogene mit quadratischer Form”. He showed that the space of all square-integrable functions on a measure space is complete under the norm given by \(\| f \|^2 = \int |f|^2 dm\).
Conclusion
A Hilbert space is a fundamental concept in mathematics and physics, used to describe spaces with complex properties. Its completeness property ensures that every Cauchy sequence converges to an element in the space, making it possible to use the inner product to represent physical quantities such as energy and momentum.