542 Chapter 9 • Sequences and Series of Functions
On this set :F(S, JR) we define algebraic operations. For every pair of functions
f,g E :F(S, JR), and Vr E JR, we define
(a) Addition off and g by specifying that
Vx ES, (! + g)(x) = f(x) + g(x).
(b) Multiplication of f by a "scalar" r by specifying that
Vx ES, (r f)(x) = r · f(x).
( c) Multiplication of f and g by specifying that
Vx ES, (fg)(x) = f(x) · g(x).
(d) Division of f by g by specifying that
(
f) f(x)
Vx ES, g (x) = g(x).(Notice that f E :F(S, JR) only if Vx ES, g(x) =I-0.)
(e) The absolute value off: Vx ES, lfl(x) = lf(x)j.
(f) The maximum off and g:
Vx ES, max{f,g}(x) = max{f(x),g(x)}.
(g) The minimum of f and g:
Vx ES, min{f, g}(x) = min{f(x),g(x)}.
Theorem 9.1.2 (Algebra of Functions) If S is an arbitrary nonempty set
then :F(S, JR), together with th e operations (a) and (b) specified in Definition
9.1.1 above, satisfies the following ten properties:
(1) Vf,g E :F(S, JR), f + g E :F(S, JR);
(2) Vf,g, h E :F(S, JR), f + (g + h) = (! + g) + h ;
(3) Vf,gE:F(S,JR), f+g=g+f;
(4) :3 0 E :F(S,JR) 3 VJ E :F(S, JR), f + 0 = 0 + f = f ;
(5) \ff E :F(S, JR), :3-f E :F(S, JR) 3 f + (-!) = O;
(6) VJ E :F(S, JR), r E JR, rf E :F(S, JR);
(7) Vf,g E :F(S, JR), V r E JR, r(f + g) = rf +rg;
(8) VJ E :F(S, JR), V r,s E JR, (r + s) (!) = rf + s f;