Every complex vector space can also been seen as real vector space. If we now choose a real subspace, it may not be a complex subspace (in particular, if it is of odd real dimension).
If the complex vector space was equipped with an inner product (for example, a Hilbert space), we can restrict the imaginary unit (also known as linear complex structure) to any real subspace using the orthogonal projection. We can then classify the types of real subspaces based on the spectrum of this ``restricted complex structure''. In particular, if the restricted complex structure squares to minus identity, i.e., is itself a complex structure, the real subspace is also a complex subspace. In general, the spectrum encodes how being a complex subspace is violated.
I worked this out for myself, but I'm confident that this is standard material in linear algebra of complex vector spaces. However, the standard introductory text books that I checked do not discuss real subspaces of complex vector spaces and their classification in the above way.
Do you know of a standard reference that I could cite when discussing this (in particular, the above mentioned classification based on the spectrum of the restricted complex structure)?
