Canonical static concept record

Complex vector space

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Summary

A vector space whose scalar field is $\mathbb C$ .

Record metadata

Canonical name: Complex vector space
Concept ID: complex_vector_space
Content version: 2.3.0
Primary field: Mathematics
Fields: Mathematics
Primary domain: Algebra
Domains: Algebra, Analysis
Alternate terminology: No alternate terminology is currently authored for this record.
Concept type: Algebraic structure

Carrier(s)

set $V$
scalar field $\mathbb C$

Data

vector addition
complex scalar multiplication

Axioms / constraints

vector-space axioms over $\mathbb C$

Canonically induces

underlying real vector space
complex-linear maps

Concept sources

Wikipedia — Complex vector space — Complex vector space · encyclopedic reference · source ID wp-complex_vector_space
nLab — complex vector space — complex vector space · mathematical reference · source ID nlab-complex_vector_space

Incoming relations (arrows to this concept)

Each relation below ends at this concept.

Vector space Complex vector space

Permalink to relation

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

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

Authored explanation

A complex vector space is a vector space over $\mathbb C$ .

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 — Complex vector space — Complex vector space · encyclopedic reference · source ID wp-complex_vector_space
nLab — complex vector space — complex vector space · mathematical reference · source ID nlab-complex_vector_space

Outgoing relations (arrows from this concept)

Each relation below starts at this concept.

Complex vector space Quantum mechanics

Permalink to relation

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

Relation ID: complexvs_to_qm
Relation type: Mathematical formulation mathematical-formulation
Direction: source → target
Endpoint roles: source: Mathematically formulates; target: Mathematically formulated using
Authored annotation: state spaces and amplitudes

Authored explanation

Quantum mechanics is ordinarily formulated over complex vector spaces.

How to interpret this relation type

A mathematical concept supplies part of the formal language, state space, representation, or analytic machinery used by a scientific or mathematical-physics concept. The source is the mathematical predecessor; this does not claim that physical content follows from mathematics alone.

Relation sources

Wikipedia — Quantum mechanics — Quantum mechanics · encyclopedic reference · source ID wp-physics-quantum_mechanics
nLab — quantum mechanics — quantum mechanics · mathematical physics reference · source ID nlab-physics-quantum_mechanics
Wikipedia — Complex vector space — Complex vector space · encyclopedic reference · source ID wp-complex_vector_space
nLab — complex vector space — complex vector space · mathematical reference · source ID nlab-complex_vector_space

Complex vector space Quantum superposition

Permalink to relation

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

Relation ID: complexvs_to_superposition
Relation type: Mathematical formulation mathematical-formulation
Direction: source → target
Endpoint roles: source: Mathematically formulates; target: Mathematically formulated using
Authored annotation: linear combinations

Authored explanation

The linear structure of complex state spaces supplies the formal superposition operation.

How to interpret this relation type

A mathematical concept supplies part of the formal language, state space, representation, or analytic machinery used by a scientific or mathematical-physics concept. The source is the mathematical predecessor; this does not claim that physical content follows from mathematics alone.

Relation sources

Wikipedia — Complex vector space — Complex vector space · encyclopedic reference · source ID wp-complex_vector_space
nLab — complex vector space — complex vector space · mathematical reference · source ID nlab-complex_vector_space
Wikipedia — Quantum superposition — Quantum superposition · encyclopedic reference · source ID wp-physics-quantum_superposition
nLab — superposition principle — superposition principle · mathematical physics reference · source ID nlab-physics-quantum_superposition
NIST — Quantum Physics Theory — Quantum Physics Theory · NIST research reference · source ID nist-quantum-theory
Cambridge — Quantum Mechanics — Quantum Mechanics · University of Cambridge lecture notes · source ID tong-quantum