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Construction junctions are diamonds. They show where multiple structures must coexist on the same carrier and satisfy compatibility conditions.

Canonical static concept record

Topological vector space

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Summary

A vector space with a compatible topology.

Record metadata

Canonical name: Topological vector space
Concept ID: topological_vector_space
Content version: 2.3.0
Primary field: Mathematics
Fields: Mathematics
Primary domain: Analysis
Domains: Analysis, Topology, Algebra
Alternate terminology: No alternate terminology is currently authored for this record.

Carrier(s)

topological field $F$
set $V$

Data

vector-space structure
topology on $V$

Axioms / constraints

Hausdorff topology
continuous vector addition
continuous scalar multiplication

Notes

This atlas adopts the common Hausdorff convention.

Concept sources

Wikipedia — Topological vector space — Topological vector space · expository · source ID wp-top-vector
nLab — Topological vector space — Topological vector space · reference · source ID nlab-top-vector
Princeton — Functional Analysis Lecture Notes — Functional Analysis Lecture Notes · course notes · source ID princeton-functional-analysis

Incoming relations (arrows to this concept)

Each relation below ends at this concept.

linear + topological compatibility(construction junction)Topological vector space

Permalink to relation

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

Relation ID: e_jtvs_tvs
Relation type: Combine compatible structures combine-compatible
Direction: source → target
Endpoint roles: source: Builds toward; target: Built from
Authored annotation: + continuity

Authored explanation

Require addition and scalar multiplication to be continuous.

How to interpret this relation type

Feed several structures into a construction junction and impose compatibility between them.

Relation sources

Wikipedia — Topological vector space — Topological vector space · expository · source ID wp-top-vector
nLab — Topological vector space — Topological vector space · reference · source ID nlab-top-vector
Princeton — Functional Analysis Lecture Notes — Functional Analysis Lecture Notes · course notes · source ID princeton-functional-analysis

Normed vector space Topological vector space

Permalink to relation

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

Relation ID: normed_to_topological_vector_space
Relation type: Induced / forgotten induced-forgotten
Direction: source → target
Endpoint roles: source: Yields by induction / forgetting; target: Obtained by induction / forgetting from
Authored annotation: induce norm topology

Authored explanation

The norm metric makes vector addition and scalar multiplication continuous.

How to interpret this relation type

Pass canonically from a stronger object to structure it determines, or forget part of the data while retaining a valid weaker structure. The carrier may change under a canonical induced construction.

Relation sources

Wikipedia — Normed vector space — Normed vector space · expository · source ID wp-normed
nLab — Norm — Norm · reference · source ID nlab-norm
Wikipedia — Topological vector space — Topological vector space · expository · source ID wp-top-vector
nLab — Topological vector space — Topological vector space · reference · source ID nlab-top-vector
Princeton — Functional Analysis Lecture Notes — Functional Analysis Lecture Notes · course notes · source ID princeton-functional-analysis

Outgoing relations (arrows from this concept)

Each relation below starts at this concept.