9.1 Families of Functions and Pointwise Convergence 543(9) VJ E F(S, JR), V r,s E JR, r(sJ) = (rs)J = s(rJ);
(10) VJ E F(S, JR), lJ = J;
Taking into account operation (c) of Definition 9.1.1, F(S, JR) also satisfies the
fallowing five properties:(11) Vf, g E F(S, JR), Jg E F(S, JR);
(12) VJ,g,h E F(S, JR), f(gh) = (fg)h;
(13) Vf,g E F(S, IR), Jg= gf;
(14) Vf,g, h E F(S, JR), f(g + h) =Jg+ fh;
(15) VJ,g E F(S, JR), andVr E JR, r(fg) = (rf)g = f(rg);
Proof. See Theorem B.3.3 in Appendix B. •Definition 9.1.3 Because F(S, JR), together with the operations of addition
and multiplication by scalars, satisfies the first ten properties of Theorem 9.1.2,
it is called a vector space^1 of functions, or simply a function space. Any
subset Q s;:; F(S, JR) that also satisfies these ten properties relative to these two
operations is called a subspace of F(S, JR).
Because F(S, JR), together with the operations of addition, multiplication
by scalars, and multiplication satisfies all fifteen properties of Theorem 9.1.2,
it is called an algebra of functions. Any subset Q s;:; F(S, JR) that also satisfies
these fifteen properties relative to these three operations is called a subalgebra
of F(S, JR).In the remainder of this chapter we shall be concerned with vector spaces
of functions and their subspaces, but we shall not pursue algebras of functions
and their subalgebras.
Lemma 9.1.4 Any subset of F(S, JR) that satisfies properties (1) and (6) of
Theorem 9.1.2 is a subspace of F(S, JR).
Proof. Consult any elementary linear algebra textbook. •- See also Theorem 8.5.11.