##  [Hilbert Space](/hilbert-space) 

  ##  [Hilbert Space](https://quantummech.quantumdictionary.io/hilbert-space-1) 

  

 [![Quantum Mechanics Dictionary](/sites/default/files/styles/large/public/2026-01/Quantum%20Mechanics.png.webp?itok=tC_FGCW0)](/topic-specific-dictionaries/modern-physics/quantum-mechanics)

- Modern Physics -

**Quantum Mechanics Dictionary**

 







 

 

 

 



 

 

 

 

Definition

A mathematical structure used to represent quantum states and the operations performed on them. It specifies how states are expressed, related, combined, and decomposed into components that support calculation. It does not by itself determine physical predictions without a mapping to observables and measurement rules.



 

 

 

 

 





 

 



 ##  [Hilbert Space](https://natural.quantumdictionary.io/hilbert-space-0) 

  

 [![Natural & Formal Sciences Dictionary](/sites/default/files/styles/large/public/2026-01/Natural%20%26%20Formal%20Sciences.png.webp?itok=2kCDRVQv)](/topic-specific-dictionaries/natural-formal-sciences)



**Natural & Formal Sciences Dictionary**

 







 

 

 

 



 

 

 

 

Definition

A complete vector space over the real or complex numbers equipped with an inner product whose induced norm makes the space complete for Cauchy sequences; supports orthogonality, projections, and unitary evolution operators.

 

 

 

 

 





 

 



 ##  [Hilbert Space](https://mathlogic.quantumdictionary.io/hilbert-space-2) 

  

 [![Mathematics & Logic Dictionary](/sites/default/files/styles/large/public/2026-01/Mathematics%20%26%20Logic.png.webp?itok=UhtTRPnp)](/topic-specific-dictionaries/natural-formal-sciences/mathematics-logic)

- Natural & Formal Sciences -

**Mathematics & Logic Dictionary**

 







 

 

 

 



 

 

 

 

Definition

A complete inner-product space (real or complex) in which the inner product induces the norm and where orthogonality, orthonormal bases, projections, and spectral methods are well-defined.

 

 

 

 

 





 

 



 ##  [Hilbert Space](https://puremath.quantumdictionary.io/hilbert-space-3) 

  

 [![Pure Mathematics Dictionary](/sites/default/files/styles/large/public/2026-01/Pure%20Mathematics.png.webp?itok=5pZnFQ59)](/topic-specific-dictionaries/mathematics-logic/pure-mathematics)

- Mathematics & Logic -

**Pure Mathematics Dictionary**

 







 

 

 

 



 

 

 

 

Definition

A vector space endowed with an inner product that is complete in the metric induced by that inner product; equivalently, a complete inner product space where the inner product induces the norm and topology.