Editing Symplectic vector space (section)

=== Lagrangian form ===
There is another way to interpret this standard symplectic form. Since the model space '''R'''<sup>2''n''</sup> used above carries much canonical structure which might easily lead to misinterpretation, we will use "anonymous" vector spaces instead. Let ''V'' be a real vector space of dimension ''n'' and ''V''<sup>∗</sup> its [[dual space]]. Now consider the [[direct sum of vector spaces|direct sum]] {{nowrap|1=''W'' = ''V'' ⊕ ''V''<sup>∗</sup>}} of these spaces equipped with the following form:

:<math>\omega(x \oplus \eta, y \oplus \xi) = \xi(x) - \eta(y).</math>

Now choose any [[Basis (linear algebra)|basis]] {{nowrap|(''v''<sub>1</sub>, ..., ''v''<sub>''n''</sub>)}} of ''V'' and consider its [[dual space|dual basis]]

:<math>\left(v^*_1, \ldots, v^*_n\right).</math>

We can interpret the basis vectors as lying in ''W'' if we write {{nowrap|1=''x''<sub>''i''</sub> = (''v''<sub>''i''</sub>, 0) and ''y''<sub>''i''</sub> = (0, ''v''<sub>''i''</sub><sup>∗</sup>)}}. Taken together, these form a complete basis of ''W'',

:<math>(x_1, \ldots, x_n, y_1, \ldots, y_n).</math>

The form ''ω'' defined here can be shown to have the same properties as in the beginning of this section. On the other hand, every symplectic structure is isomorphic to one of the form {{nowrap|''V'' ⊕ ''V''<sup>∗</sup>}}. The subspace ''V'' is not unique, and a choice of subspace ''V'' is called a '''polarization'''. The subspaces that give such an isomorphism are called '''Lagrangian subspaces''' or simply '''Lagrangians'''.

Explicitly, given a Lagrangian subspace [[#Subspaces|as defined below]], then a choice of basis {{nowrap|(''x''<sub>1</sub>, ..., ''x<sub>n</sub>'')}} defines a dual basis for a complement, by {{nowrap|1=''ω''(''x''<sub>''i''</sub>, ''y''<sub>''j''</sub>) = ''δ''<sub>''ij''</sub>}}.