Editing Symplectic vector space (section)

===Analogy with complex structures===
Just as every symplectic structure is isomorphic to one of the form {{nowrap|''V'' ⊕ ''V''<sup>∗</sup>}}, every [[linear complex structure|''complex'' structure]] on a vector space is isomorphic to one of the form {{nowrap|''V'' ⊕ ''V''}}. Using these structures, the [[tangent bundle]] of an ''n''-manifold, considered as a 2''n''-manifold, has an [[almost complex structure]], and the [[cotangent bundle|''co''tangent bundle]] of an ''n''-manifold, considered as a 2''n''-manifold, has a symplectic structure: {{nowrap|1=''T''<sub>∗</sub>(''T''<sup>∗</sup>''M'')<sub>''p''</sub> = ''T''<sub>''p''</sub>(''M'') ⊕ (''T''<sub>''p''</sub>(''M''))<sup>∗</sup>}}.

The complex analog to a Lagrangian subspace is a [[real subspace|''real'' subspace]], a subspace whose [[complexification]] is the whole space: {{nowrap|1=''W'' = ''V'' ⊕ ''J'' ''V''}}. As can be seen from the standard symplectic form above, every symplectic form on '''R'''<sup>2''n''</sup> is isomorphic to the imaginary part of the standard complex (Hermitian) inner product on '''C'''<sup>''n''</sup> (with the convention of the first argument being anti-linear).