“real: chapter_08” — 2011/5/22 — 23:34 — page 18 — #18
8-18 Real Analysis
8.5 PROPERTIES OF RIEMANN INTEGRALS
Theorem 8.5.1 The set R[a, b] is a real vector space and the map
:
R[a, b]→R given by (f ) =
b
a
f (x)dx is a linear functional
(linear map with values in the base field
R).
Proof We have to show that if f , g ∈
R[a, b] and α, β ∈ R, then
αf + βg ∈
R[a, b] and that
(αf + βg) = α(f ) + β(g)
or equivalently
b
a
(αf (x) + βg(x))dx = α
b
a
f (x)dx + β
b
a
g(x)dx.
Here (f +g)(x) = f (x)+g(x) and (αf )(x) = α(f (x)) for all x ∈[a, b].
We shall use Theorem 8.4.2 to complete the proof.
Let >0 be given. Choose η>0 such that (|α|+|β|)η < . Since
f , g ∈
R[a, b], there are partitions P
1
and P
2
of [a, b] such that for any
refinements Q
i
of P
i
(i = 1, 2), we have