A _Hilbert space_, $\mathcal{H},$ is a [complex vector space](Complex%20vector%20spaces.md) with the following properties: 1) It allows for an [Inner product](Inner%20products.md) between any two vector elements. ^7a298d 2) There may be either a finite [$n$](Finite%20dimensional%20Hilbert%20spaces.md) or [$\infty$ ](Infinite%20dimensional%20Hilbert%20spaces.md)_linearly independent_ vectors in $\mathcal{H}$ (this property is implied by the fact that it is a linear vector space with an inner product). ^c27d11 3) $\mathcal{H}$ is [complete sequence](Complete%20sequence.md). Note completeness can be assumed if $\mathcal{H}$ is finite dimensional and must be [proven if infinite dimensional.](Infinite%20dimensional%20Hilbert%20spaces.md#Proof%20that%20infinite%20dimensional%20Hilbert%20Spaces%20are%20complete) ^8cb949 Thus a Hilbert space is defined as any _complete_ [complex inner product space.](Complex%20vector%20spaces.md#Complex%20inner%20product%20spaces) Being complex inner product space, this means we may define an inner product for vectors $|u\rangle,|v\rangle\in\mathcal{H},$ such that [$\langle x|y \rangle = \sum_j x^*_j y_j$](Complex%20vector%20spaces.md#^363959) where we will use the [vector notation](Linear%20Algebra%20and%20Matrix%20Theory%20(index).md#Vector%20notation) as defined for vectors in Hilbert spaces. %%The completeness of finite dimensional hilbert spaces still must be elaborated on. This is a point that's trivial to mathematician but not trivial to math students.%% ## Separable and non-Separable Hilbert Spaces In von Neumann's mathematical formulation of quantum mechanics the Hilbert space is also required to be _separable_ ^[[von Neumann J., _Mathematical Foundations of Quantum Mechanics_. Translated by Robert T. Beyer. Princeton University Press, 2018.](von%20Neumann%20J.,%20Mathematical%20Foundations%20of%20Quantum%20Mechanics.%20Translated%20by%20Robert%20T.%20Beyer.%20Princeton%20University%20Press,%202018..md)]. i.e. $\mathcal{H}$ must have a _countable_ orthonormal basis. This is not true for all Hilbert Spaces. In general a Hilbert Space is non-separable if it is composed of an infinite [tensor product](Tensor%20product%20of%20Hilbert%20Spaces.md) of separable Hilbert spaces. # Generalization of Euclidean Space For [Hilbert spaces](Hilbert%20Space.md) the properties of [Euclidian spaces](Euclidean%20space.md) are generalized to [n](Finite%20dimensional%20Hilbert%20spaces.md) or [infinite](Infinite%20dimensional%20Hilbert%20spaces.md) dimensions on $\mathbb{C}.$ # Orthonormal Bases [Orthonormal bases](Orthonormal%20bases.md) %%You need to prove at some point that a Hilbert space always implies an orthonormal basis%% # Hilbert Space Operators --- # Proofs and Examples --- # Recommended Reading For a thorough and formal definition of the [[Hilbert Space]] see: * [von Neumann J., _Mathematical Foundations of Quantum Mechanics_. Translated by Robert T. Beyer. Princeton University Press, 2018.](von%20Neumann%20J.,%20Mathematical%20Foundations%20of%20Quantum%20Mechanics.%20Translated%20by%20Robert%20T.%20Beyer.%20Princeton%20University%20Press,%202018..md) pgs. 25-32. The definition of the Hilbert Space used here most closely follows the one presented here. In addition this text includes a lengthy chapter on the properties of Hilbert Spaces and what we can do with them on pgs. 25 to 126. For a surface-level discussion of the [separability](Hilbert%20Space.md#Separable%20and%20non-Separable%20Hilbert%20Spaces) of Hilbert Spaces and its connection to [[Hilbert Spaces in Quantum Mechanics]] see: * [Streater R. F., Wightman A. S. _PCT Spin and Statistics and All That_, Princeton University Press, 2000]([Princeton%20Landmarks%20In%20Mathematics%20And%20Physics]%20Raymond%20F.%20Streater,%20Arthur%20S.%20Wightman%20-%20PCT,%20Spin%20And%20Statistics,%20And%20All%20That%20(2000,%20Princeton%20University%20Press)%20-%20libgen.lc.pdf) pgs. 85-87. This text is aimed at graduate students studying theoretical or mathematical physics. #MathematicalFoundations/Algebra/AbstractAlgebra/LinearAlgebra/VectorSpaces #MathematicalFoundations/Analysis/FunctionalAnalysis