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Canonical static concept record

Densely defined operator

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Summary

A linear operator defined on a dense linear subspace of a topological vector space, commonly a Hilbert space.

Record metadata

Canonical name: Densely defined operator
Concept ID: densely_defined_operator
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: Possibly unbounded operator

Carrier(s)

Hilbert or Banach space $H$
dense linear domain $D(A)\subset H$

Data

linear map $A:D(A)\to H$

Axioms / constraints

linearity on its domain

Canonically induces

adjoint when the setting is Hilbertian
closure if closable

Notes

An unbounded operator cannot be defined and bounded on all of a Banach space; its domain is part of the operator.

Concept sources

Wikipedia — Unbounded operator — Unbounded operator · encyclopedic reference · source ID wp-densely_defined_operator
nLab — unbounded operator — unbounded operator · mathematical reference · source ID nlab-densely_defined_operator

Incoming relations (arrows to this concept)

Each relation below ends at this concept.

Linear operator Densely defined operator

Permalink to relation

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

Relation ID: math_linear_operator_to_densely_defined_operator
Relation type: Framework specialization framework-specialization
Direction: source → target
Endpoint roles: source: Specializes to; target: Special case or refinement of
Authored annotation: dense domain

Authored explanation

A densely defined operator may be unbounded and is specified on a dense subspace.

How to interpret this relation type

A more specific theory, model, entity class, or regime is obtained by restricting or extending the scope of a broader framework.

Relation sources

Wikipedia — Unbounded operator — Unbounded operator · encyclopedic reference · source ID wp-densely_defined_operator
nLab — unbounded operator — unbounded operator · mathematical reference · source ID nlab-densely_defined_operator

Outgoing relations (arrows from this concept)

Each relation below starts at this concept.