Answers & Hints for Selected Exercises 707- Let J, g E F(S, ~).Then, Vx ES, (! + g)(x) = J(x) + g(x) = g(x) + J(x) =
(g + f)(x) and (f g)(x ) = J(x)g(x) = g(x)J(x) = (gf)(x ). .'. J + g = g + J and
Jg= gf.
7. Let J,g E F(S, ~).Then, Vx ES, r(f + g) = r[(f + g)(x)] = r[J(x) +
g(x)] = rJ(x) +rg(x) = (r f)(x) + (rg)(x) = (rJ +rg)(x). .'. r(f + g) = r J +rg.
9. Let J E F(S, ~). Then, Vx ES, (lf)(x) = 1 · J(x) = J(x). .'. lJ = l.
11. Let J, g, h E F(S, ~). Then, Vx E S, J(g + h) = J(x)(g + h)(x) =
J(x)[g(x) + h(x)] = J(x)g(x) + J(x)h(x) = (fg)(x) + (fh)(x) =(Jg+ Jh)(x ).
.'. J(g + h) =Jg+ Jh.
13. J(x) = x^2 , g(x) = x^3.
15. (a) Va EA, (! o iA)(a) = J (iA(a)) = J(a), so Jo iA = f.
(b) Vb EB, (iA 0 g)(b) = iA (g(b)) = g(b), so iA 0 g = g.