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

How statement and proof provenance work

The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.

Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

These labels describe origin, not correctness: citations and verification chips remain separate evidence.

Isotropic, coisotropic, symplectic, and Lagrangian subspaces

Definition

Let $W$ be a subspace of a symplectic vector space $(V, ω)$ , with symplectic orthogonal $W^{ω}$ . Then $W$ is

isotropic when $W \subseteq W^{ω}$ , equivalently $ω ∣_{W \times W} = 0$ ;
coisotropic when $W^{ω} \subseteq W$ ;
symplectic when $ω ∣_{W \times W}$ is nondegenerate, equivalently $W \cap W^{ω} = 0$ ; and
Lagrangian when $W = W^{ω}$ .

The zero subspace is isotropic (and Lagrangian only when $V = 0$ ), while $V$ is coisotropic and symplectic.

Depends on

Symplectic orthogonal complement

Used by

Dependency tree · two levels

2 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

Eckhard Meinrenken, Symplectic Geometry (standard reference, not scraped)