Lang 2002, p. 142 "Let E be a free module over a commutative ring A. We view A as a free module of rank 1 over itself. By the dual moduleE∨ of E we shall mean the module Hom(E, A). Its elements will be called functionals. Thus a functional on E is an A-linear map f : E → A." Lang, Serge (2002), "III. Modules, §6. The dual space and dual module", Algebra, Graduate Texts in Mathematics, vol. 211 (Revised third ed.), New York: Springer-Verlag, pp. 142–146, ISBN978-0-387-95385-4, MR1878556, Zbl0984.00001
Kolmogorov & Fomin 1957, p. 77 "A numerical function f(x) defined on a normed linear space R will be called a functional. A functional f(x) is said to be linear if f(αx + βy) = αf(x) + βf(y) where x, y ∈ R and α, β are arbitrary numbers." Kolmogorov, Andrey; Fomin, Sergei V. (1957). Elements of the Theory of Functions and Functional Analysis. Dover Books on Mathematics. New York: Dover Books. ISBN978-1-61427-304-2. OCLC912495626.
Kolmogorov & Fomin 1957, pp. 62-63 "A real function on a space R is a mapping of R into the space R1 (the real line). Thus, for example, a mapping of Rn into R1 is an ordinary real-valued function of n variables. In the case where the space R itself consists of functions, the functions of the elements of R are usually called functionals." Kolmogorov, Andrey; Fomin, Sergei V. (1957). Elements of the Theory of Functions and Functional Analysis. Dover Books on Mathematics. New York: Dover Books. ISBN978-1-61427-304-2. OCLC912495626.
zbmath.org
Lang 2002, p. 142 "Let E be a free module over a commutative ring A. We view A as a free module of rank 1 over itself. By the dual moduleE∨ of E we shall mean the module Hom(E, A). Its elements will be called functionals. Thus a functional on E is an A-linear map f : E → A." Lang, Serge (2002), "III. Modules, §6. The dual space and dual module", Algebra, Graduate Texts in Mathematics, vol. 211 (Revised third ed.), New York: Springer-Verlag, pp. 142–146, ISBN978-0-387-95385-4, MR1878556, Zbl0984.00001
