A rigged Hilbert space is a pair (H,Φ) with H a Hilbert space, Φ a dense subspace, such that Φ is given a topological vector space structure for which the inclusion map i is continuous.
Identifying H with its dual space H*, the adjoint to i is the map
- .
The duality pairing between Φ and Φ* has to be compatible with the inner product on H, in the sense that:
whenever and .
The specific triple is often named the "Gelfand triple" (after the mathematician Israel Gelfand).
Note that even though Φ is isomorphic to Φ* if Φ is a Hilbert space in its own right, this isomorphism is not the same as the composition of the inclusion i with its adjoint i*
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