Editing Bra–ket notation (section)

=== Non-normalizable states and non-Hilbert spaces ===

Bra–ket notation can be used even if the vector space is not a [[Hilbert space]].

In quantum mechanics, it is common practice to write down kets which have infinite [[norm (mathematics)|norm]], i.e. non-[[normalizable wavefunction]]s. Examples include states whose wavefunctions are [[Dirac delta function]]s or infinite [[plane wave]]s. These do not, technically, belong to the [[Hilbert space]] itself. However, the definition of "Hilbert space" can be broadened to accommodate these states (see the [[Gelfand–Naimark–Segal construction]] or [[rigged Hilbert space]]s). The bra–ket notation continues to work in an analogous way in this more general context.

[[Banach spaces]] are a different generalization of Hilbert spaces. In a Banach space {{math|{{mathcal|B}}}}, the vectors may be notated by kets and the continuous [[linear functionals]] by bras. Over any vector space without a given [[topology]], we may still notate the vectors by kets and the linear functionals by bras. In these more general contexts, the bracket does not have the meaning of an inner product, because the [[Riesz representation theorem]] does not apply.