# Definition A Hilbert space $\mathcal{H}$ is a complete [[vector space]] equipped with an [[inner product]]. # In Physics Hilbert spaces are often used in quantum mechanics. Any *quantum-mechanical* Hilbert space should carry a [[Unitary Representation|unitary representation]] $\pi:\mathfrak{g} \rightarrow \mathfrak{isom}(\mathcal{H})$ of any physically relevant [[Lie algebra]] $\mathfrak{g}$. Here, $\mathfrak{isom}(\mathcal{H})$ denotes the [[isometry group|isometry Lie algebra]] (i.e. the vector space of all [[Hermitian Operator|anti-Hermitian operators]] [or Hermitian in the [[Physicists' Definition of Lie Algebras|physicists' definition]]] equipped with a [[commutator]]) on $\mathcal{H}$.