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Canonical static concept record

Direct sum of vector spaces

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Summary

The direct sum $\bigoplus_i V_i$ consists of tuples with finite support in the algebraic case; Hilbert direct sums complete square-summable families under the natural inner product.

Record metadata

Canonical name: Direct sum of vector spaces
Concept ID: direct_sum_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: Coproduct / sum construction

Carrier(s)

family of vector spaces $(V_i)_{i\in I}$

Data

finite-support tuples $(v_i)$

Axioms / constraints

componentwise operations
only finitely many nonzero components in the algebraic direct sum

Canonically induces

canonical injections and projections

Notes

The Hilbert direct sum is a different completed construction using square-summable families.

Concept sources

Wikipedia — Direct sum — Direct sum · encyclopedic reference · source ID wp-direct-sum-vector-space
nLab — direct sum — direct sum · mathematical reference · source ID nlab-direct-sum-vector-space

Incoming relations (arrows to this concept)

Each relation below ends at this concept.

Vector space Direct sum of vector spaces

Permalink to relation

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

Relation ID: vectorspace_to_directsum
Relation type: Canonical construction canonical-construction
Direction: source → target
Endpoint roles: source: Canonically constructs; target: Canonically constructed from
Authored annotation: indexed direct sum

Authored explanation

An indexed family of vector spaces has a direct sum characterized by finitely supported components and the coproduct universal property.

How to interpret this relation type

Apply a standard functorial or canonical construction whose output is not merely a reduct of the input and is not generally an equivalent presentation of the same object.

Relation sources

Wikipedia — Direct sum — Direct sum · encyclopedic reference · source ID wp-direct-sum-vector-space
nLab — direct sum — direct sum · mathematical reference · source ID nlab-direct-sum-vector-space

Outgoing relations (arrows from this concept)

Each relation below starts at this concept.