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1-5 Preliminaries

Elsevier B.V.
DOI: 10.1016/s0079-8169(08)62106-6
  • Mathematics


Publisher Summary This chapter discusses the complex extension, real restriction, and waiving processes. Linear spaces (Spa Lin) will in general be taken over the fields of real and complex numbers as scalars (Spa Lin Rea and Spa Lin Com). Algebras will be considered over the same fields (Alg Rea and Alg Com). A complex extension of R ∈ Spa Lin Rea is an S ∈ Spa Lin Com which as a real linear space is the direct sum of R and iR. Two complex extensions of R are canonically isomorphic. In the text R will often occur as a real subspace of a complex linear space, with R∩ iR = {0}, and then the complex span of R is a complex extension of R, usually denoted by Rcom and referred to as the complex extension of R. In other cases it makes sense to reserve this notation and terminology for the complex extension of R obtained by means of the tensor product of R and Com. By ignoring scalar multiplication with nonreals, Spa Lin Com is mapped into Spa Lin Rea. This mapping is called “waiving.” By complex extension and real restriction the dimension is preserved; by waiving it doubles. A topological group (Gru Top) is a set with a group structure and a topology such that multiplication and inversion are continuous operations in the given topology.

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