Mediaspace scheduled maintenance: Aug 25, 2026 07:00 - 12:00 AM. During this time, videos will be temporarily unavailable. Check status updates.

Lecture

Orthonormal Sequences and Hilbert Spaces

In course

MATH-495: Mathematical quantum mechanics

Quantum mechanics is one of the most successful physical theories. This course presents the mathematical formalism (functional analysis and spectral theory) that underlies quantum mechanics. It is simultaneously an introduction to mathematical physics and ...

Description

This lecture covers the concept of orthonormal sequences in Hilbert spaces, explaining how they can be used to decompose elements into orthogonal components. It also delves into Bessel's inequality, Cauchy-Schwarz inequality, and the importance of separable Hilbert spaces. The instructor discusses the relationship between orthonormal bases and algebraic bases, the polarization identity, and the characterization of orthonormal bases. The lecture concludes with a detailed proof involving orthogonal complements and unique decompositions in Hilbert spaces.

This video is available exclusively on Mediaspace for a restricted audience. Please log in to MediaSpace to access it if you have the necessary permissions.

Watch on Mediaspace

Official source

Ontological neighbourhood

Mathematics

Algebra: Linear algebra

Related concepts (32)

Hilbert space

In mathematics, Hilbert spaces (named after David Hilbert) allow the methods of linear algebra and calculus to be generalized from (finite-dimensional) Euclidean vector spaces to spaces that may be infinite-dimensional. Hilbert spaces arise naturally and frequently in mathematics and physics, typically as function spaces. Formally, a Hilbert space is a vector space equipped with an inner product that induces a distance function for which the space is a complete metric space.

Reproducing kernel Hilbert space

In functional analysis (a branch of mathematics), a reproducing kernel Hilbert space (RKHS) is a Hilbert space of functions in which point evaluation is a continuous linear functional. Roughly speaking, this means that if two functions and in the RKHS are close in norm, i.e., is small, then and are also pointwise close, i.e., is small for all . The converse does not need to be true. Informally, this can be shown by looking at the supremum norm: the sequence of functions converges pointwise, but does not converge uniformly i.

Hilbert cube

In mathematics, the Hilbert cube, named after David Hilbert, is a topological space that provides an instructive example of some ideas in topology. Furthermore, many interesting topological spaces can be embedded in the Hilbert cube; that is, can be viewed as subspaces of the Hilbert cube (see below).

Orthonormal basis

In mathematics, particularly linear algebra, an orthonormal basis for an inner product space V with finite dimension is a basis for whose vectors are orthonormal, that is, they are all unit vectors and orthogonal to each other. For example, the standard basis for a Euclidean space is an orthonormal basis, where the relevant inner product is the dot product of vectors. The of the standard basis under a rotation or reflection (or any orthogonal transformation) is also orthonormal, and every orthonormal basis for arises in this fashion.

Matrix decomposition

In the mathematical discipline of linear algebra, a matrix decomposition or matrix factorization is a factorization of a matrix into a product of matrices. There are many different matrix decompositions; each finds use among a particular class of problems. In numerical analysis, different decompositions are used to implement efficient matrix algorithms. For instance, when solving a system of linear equations , the matrix A can be decomposed via the LU decomposition.

Related lectures (35)

Dirac's Notation, Tensor Product

Covers Dirac's notation in linear algebra and the tensor product concept in Hilbert spaces.

Orthogonal/Orthonormal Bases and Polynomials

Explores orthogonal and orthonormal bases, Gram-Schmidt process, and orthogonal polynomials in physics.

Orthogonal Projections in Linear Algebra

Explores orthogonal projections, orthonormal bases, and QR factorization in linear algebra.

Quantization: Topological Operators

Covers the quantization of topological operators and Ising models on square lattices.

Complete and Orthonormal Families