Subsets and subspaces

Several rules known to be valid within the set theory are revisited and considered from the point of view of the question whether they remain satisfied when sets are replaced with subspaces from Cn,1Cn,1. It is shown that some of the set relationships hold true upon such a replacement, however, there is also an extensive class of relationships whose validity is no longer guaranteed; in this class the commutativity of the orthogonal projectors onto the involved subspaces proves to play an important role. Some related results inspired by probability theory are established as well.