In mathematics, the Wirtinger inequality, named after Wilhelm Wirtinger, is a fundamental result in complex linear algebra which relates the symplectic and volume forms of a hermitian inner product.
[1] In the language of the comass of a form, the Wirtinger theorem (although without precision about when equality is achieved) can also be phrased as saying that the comass of the form ω ∧ ⋅⋅⋅ ∧ ω is equal to k!.
Then there is an orthonormal basis e1, ..., e2k of T with dual basis w1, ..., w2k such that where ι denotes the inclusion map from T into V.[2] This implies which in turn implies where the inequality follows from the previously-established k = 1 case.
This is equivalent to either ω(e2i − 1, e2i) = 1 or ω(e2i, e2i − 1) = 1, which in either case (from the k = 1 case) implies that the span of e2i − 1, e2i is closed under J, and hence that the span of e1, ..., e2k is closed under J.
Using the Wirtinger inequality, these facts even extend to the more sophisticated context of currents in Kähler manifolds.