Abstract.

Linear spaces are considered in the following four situations: a real space admits multiplication by complex scalars without changing the set itself; a real space is embedded into a wider set with multiplication by complex scalars; a complex space has an involution and thus has “real” and “imaginary” elements; a combination of the previous ones. We study how they manifest themselves when the initial space possesses additional structures such as topology, norm, inner product, and also the behavior of linear operators between such spaces.