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Canonical static concept record

Rigged Hilbert space

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Summary

A dense test-function space $\Phi$ embedded in a Hilbert space $H$ , embedded in the continuous dual $\Phi^*$ .

Record metadata

Canonical name: Rigged Hilbert space
Concept ID: rigged_hilbert_space
Content version: 2.3.0
Primary field: Mathematics
Fields: Mathematics
Primary domain: Analysis
Domains: Analysis
Alternate terminology: No alternate terminology is currently authored for this record.
Concept type: Functional-analytic triple

Carrier(s)

locally convex test space $\Phi$
Hilbert space $H$
continuous dual $\Phi^*$

Data

continuous dense embeddings $\Phi\subset H\subset\Phi^*$

Axioms / constraints

compatibility of embeddings and dual pairing

Canonically induces

generalized eigenvectors as elements of $\Phi^*$

Notes

Often called a Gelfand triple; nuclearity is common but not part of every definition.

Concept sources

Wikipedia — Rigged Hilbert space — Rigged Hilbert space · encyclopedic reference · source ID wp-rigged_hilbert_space
nLab — rigged Hilbert space — rigged Hilbert space · mathematical reference · source ID nlab-rigged_hilbert_space

Incoming relations (arrows to this concept)

Each relation below ends at this concept.

Hilbert space Rigged Hilbert space

Permalink to relation

This is an authored directed relation from the source endpoint to the target endpoint.

Relation ID: math_hilbert_space_to_rigged_hilbert_space
Relation type: Add data add-data
Direction: source → target
Endpoint roles: source: Builds toward; target: Built from
Authored annotation: test-space embedding

Authored explanation

A rigged Hilbert space chooses a dense nuclear or locally convex test space and its dual.

How to interpret this relation type

Equip an existing carrier or structured object with additional chosen data, when such compatible data exists. Use a construction junction when several independently meaningful inputs must coexist on the same carrier or interact compatibly.

Relation sources

Wikipedia — Rigged Hilbert space — Rigged Hilbert space · encyclopedic reference · source ID wp-rigged_hilbert_space
nLab — rigged Hilbert space — rigged Hilbert space · mathematical reference · source ID nlab-rigged_hilbert_space

Outgoing relations (arrows from this concept)

Each relation below starts at this concept.