In quantum mechanics the [Hilbert Space](Hilbert%20Space.md) is any [L2(I) space](L2(I)%20space.md) containing the [state vectors](State%20vector.md) that model quantum systems.
%%Wait no, are finite dimensional hilbert spaces in qm also L2 spaces?%%
# 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:
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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%%
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# 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