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. %%Note, do not confuse this notion of a non separable hilbert space with the notion of a hilbert space that is rigged. thats different.%% # 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 $\mathbb{C}.$ # Connection between Hilbert Spaces and [$L^2(I)$ space](L2(I)%20space.md) spaces %%The connection between L2 spaces and Hilbert Spaces seems somewhat technical and needs elaborating. It seems like often times they are the same thing or usually there are the same but there are a few constraints here.%% # 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 # Rigged Hilbert Spaces ([... see more](Rigged%20Hilbert%20space)) --- # 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