One of the most important results of functional analysis is that for every bounded linear functional f: H → ℂ on a Hilbert space H, there exists a fixed |v> in H such that f(|u>) is equal to the inner product of |v> with |u> for all |u> in H. This justifies the labeling of f as <v| in the bra-ket notation used in quantum mechanics.(adsbygoogle = window.adsbygoogle || []).push({});

I want to know the relationship of bra's to inner products carries over to Rigged Hilbert Space. To construct the Rigged Hilbert Space, we start with a Hilbert space H and pick out a nuclear subspace [itex]\Phi[/itex], a dense subspace of H on which certain unbounded linear operators on H are defined. The space [itex]\Phi^{\times}[/itex] of continuous anti-linear functionals will be our ket space, and the space [itex]\Phi'[/itex] of continuous linear functionals will be our bra space. We can then define a distribution-valued inner product on [itex]\Phi^{\times}[/itex]. Also, we can define a conjugate map between [itex]\Phi'[/itex] and [itex]\Phi^{\times}[/itex], so that for every |ψ> in the ket space there corresponds a bra <ψ| and vice versa.

Now my question is, can the action of an arbitrary bra <ψ| on the ket space be equated with the action of the distribution-valued inner product with |ψ>?

Any help would be greatly appreciated.

Thank You in Advance.

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# Do bras and inner products relate in a Rigged Hilbert Space?

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