In quantum mechanics the [Hilbert Space](Hilbert%20Space.md) is any [$L^2(I)$ space](L2(I)%20space.md) containing the [state vectors](State%20vector.md) that model [discrete eigenstate quantum systems.](Discrete%20eigenstate%20quantum%20systems.md) The choice to also restrict state vectors to $L^2(I)$ is motivated by the requirement that state vectors [are normalized](state%20vector%20normalization.md) to $1.$ %%This doesn't fully explain it but points us towards one direction note that the L2 space is the only Lp space that is also a Hilbert Space.%% In order to also account for the properties of [state vectors](State%20vector.md#State%20vectors%20with%20continuous%20eigenstates) in [continuous eigenstate quantum systems](Continuous%20eigenstate%20quantum%20systems.md) %%Wait no, are finite dimensional hilbert spaces in qm also L2 spaces? in qm it seems to be the case that all hilbert spaces are L2 but it does not seem to be the case in general that L2 spaces are all hilbert spaces or vice versa but it seems to be almost always the case. Additionally this whole question of whether position and momentum state vectors are Hilbert spaces is surprisingly deep and Woit seems to implicitly force everything to be a Hilbert space, other sources such as MIT opencourseware will avoid calling them Hilbert Space elements, and other sources like Ballentine will use the notion of a rigged Hilbert space.%% # Properties of Hilbert spaces in quantum mechanics The [[Hilbert Space]], $\mathcal{H}$ containing [state vectors](State%20vector.md) that model a quantum system contains the following properties that are present for all Hilbert spaces:    ^bb8d84 In addition, regardless of the degrees of freedom for our quantum mechanical system the following property holds: 4. [Separability](Hilbert%20Space.md#Separable%20and%20non-Separable%20Hilbert%20Spaces) # Hilbert Space Dimension  [(... see more)](Hilbert%20space%20dimension%20in%20quantum%20mechanics.md) %%Here would be an interesting place to introduce quantum triviality if you knew more about it.%% ## Finite Dimensional Hilbert Spaces [(... see more)](Finite%20dimensional%20Hilbert%20spaces.md) ### 2 Dimensional Hilbert Spaces  ## Infinite Dimensional Hilbert Spaces [(... see more)](Hilbert%20space%20dimension%20in%20quantum%20mechanics.md#Infinite%20dimensional%20Hilbert%20spaces%20in%20quantum%20mechanics) # Hilbert space tensor products [$\mathcal{H}=\mathcal{H}_1 \otimes \mathcal{H}_2 ... \otimes \mathcal{H}_2.$](Tensor%20product%20of%20Hilbert%20Spaces#^18d109) [(... see more)](Tensor%20product%20of%20Hilbert%20Spaces.md) ## Non-locality of Hilbert spaces in quantum mechanics %%This comes up in discussions of entanglement. Adding this subsection was inspired by the lecture by Roman Orus at Les Houches%% --- # Recommended Reading For an elementary introduction of the role of [Hilbert Spaces](Hilbert%20Space.md) in quantum mechanics that includes a fair amount of background information see: * [Griffiths D. J., _Introduction to Quantum Mechanics_, Pearson Prentice Hall, 2nd edition, 2005.](Griffiths%20D.%20J.,%20Introduction%20to%20Quantum%20Mechanics,%20Pearson%20Prentice%20Hall,%202nd%20edition,%202005..md) pgs. 93-96 For a brief discussion of separability of Hilbert Spaces and its consequence for systems in quantum mechanics see: * [Streater R. F., Wightman A. S. _PCT Spin and Statistics and All That_, Princeton University Press, 2000, pgs. 85-87]([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) #QuantumMechanics/MathematicalFoundations