Linear Structure

Abstract

Vector spaces over R and over C have special properties which derive from the fields which underlie them. Since each such space is a union of its lines, we shall call them LINEAR SPACES, the former REAL LINEAR SPACES and the latter COMPLEX LINEAR SPACES. A vector subspace S of a linear space X will be called a LINEAR SUBSPACE of X and X will be called a LINEAR SUPERSPACE of S. Vector space homomorphisms between real linear spaces or between complex linear spaces will be called LINEAR MAPS or LINEAR OPERATORS. Algebras over ℝ will be called REAL ALGEBRAS and algebras over ℂ COMPLEX ALGEBRAS. Many of the statements we shall make regarding linear spaces are valid where either ℝ or ℂ is the field involved. We shall therefore use K to identify either ℝ or ℂ, and the reader may assume that, in any context where it appears, the symbol K may be replaced consistently by either ℝ or ℂ; where no field is mentioned, it is to be understood that either ℝ or ℂ is intended.

To draw with idle spiders’ strings Most ponderous and substantial things! Measure for Measure, lll,ii.