DefinitionDefinition: AI-adaptedProof: Not applicablePipeline-generatedprecheck passjudge pass (gpt-5.6-terra)audited 2026-09-14

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Symplectic vector space

Definition

A symplectic vector space is a pair $(V, ω)$ consisting of a finite-dimensional real vector space $V$ and an alternating bilinear form $ω : V \times V \to R$ for which

$ω^{♭} : V ⟶ V^{*}, ω^{♭} (v) = ω (v, \cdot),$

is an isomorphism. Equivalently, $ω (v, w) = 0$ for every $w \in V$ implies $v = 0$ . The equivalence also follows from the radical clause in Every alternating form on a finite-dimensional space has a basis of symplectic pairs followed by a basis of its radical; in particular its rank is even. The zero vector space, with its unique alternating form, is included: its map to its dual is the unique isomorphism.

Depends on

Every alternating form on a finite-dimensional space has a basis of symplectic pairs followed by a basis of its radical; in particular its rank is even

Used by

Compatible complex structure on a symplectic vector space Definition
Symplectic form and symplectic manifold Definition
Symplectic orthogonal complement Definition
The standard symplectic vector space Example
Symplectic vector spaces have even dimension Proposition

Dependency tree · two levels

6 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

Ana Cannas da Silva, Lectures on Symplectic Geometry (standard reference, not scraped)