Extension To Related Vector Spaces
Let V be a real vector space with a complex structure J. The dual space V* has a natural complex structure J* given by the dual (or transpose) of J. The complexification of the dual space (V*)C therefore has a natural decomposition
into the ±i eigenspaces of J*. Under the natural identification of (V*)C with (VC)* one can characterize (V*)+ as those complex linear functionals which vanish on V−. Likewise (V*)− consists of those complex linear functionals which vanish on V+.
The (complex) tensor, symmetric, and exterior algebras over VC also admit decompositions. The exterior algebra is perhaps the most important application of this decomposition. In general, if a vector space U admits a decomposition U = S ⊕ T then the exterior powers of U can be decomposed as follows:
A complex structure J on V therefore induces a decomposition
where
All exterior powers are taken over the complex numbers. So if VJ is has complex dimension n (real dimension 2n) then
The dimensions add up correctly as a consequence of Vandermonde's identity.
The space of (p,q)-forms Λp,q VJ* is the space of (complex) multilinear forms on VC which vanish on homogeneous elements unless p are from V+ and q are from V−. It is also possible to regard Λp,q VJ* as the space of real multilinear maps from VJ to C which are complex linear in p terms and conjugate-linear in q terms.
See complex differential form and almost complex manifold for applications of these ideas.
Read more about this topic: Linear Complex Structure
Famous quotes containing the words extension, related and/or spaces:
“Predatory capitalism created a complex industrial system and an advanced technology; it permitted a considerable extension of democratic practice and fostered certain liberal values, but within limits that are now being pressed and must be overcome. It is not a fit system for the mid- twentieth century.”
—Noam Chomsky (b. 1928)
“A parent who from his own childhood experience is convinced of the value of fairy tales will have no difficulty in answering his child’s questions; but an adult who thinks these tales are only a bunch of lies had better not try telling them; he won’t be able to related them in a way which would enrich the child’s life.”
—Bruno Bettelheim (20th century)
“Though there were numerous vessels at this great distance in the horizon on every side, yet the vast spaces between them, like the spaces between the stars,—far as they were distant from us, so were they from one another,—nay, some were twice as far from each other as from us,—impressed us with a sense of the immensity of the ocean, the “unfruitful ocean,” as it has been called, and we could see what proportion man and his works bear to the globe.”
—Henry David Thoreau (1817–1862)