MATH 321 - HOMEWORK #3
Due Friday, Jan 29.
The first two problems in this homework come from the textbook. See problems 10-12 at
the end of Chapter 6 for hints. Sections 11.34 – 11.38 prove the same results more generally
using the Lebesgue integral. You may adapt these proofs to the case of Riemann integrals.
An inner product space is an R-vector space together with an inner product h·, ·i that
n
satisfies properties similar
p to the dot product in R . On an inner product space one can
define the norm kvk = hv, vi and distance between two vectors d(v, w) = kv − wk.
Let us fix an interval [a, b] and a non-decreasing function α, and let R(α) be the set of
Riemann integrable functions on [a, b]. Recall that this is a vector space and it is also close
under multiplication. Define the L2 inner product on R(α) by
Zb
hf, gi2 = fgdα,
a
the L2 norm
Zb
kfk2 =
1
f2 dα 2 ,
a
and the associated distance function. The inner product and norm defined this way are not
a true inner product and norm because there exist functions f(x) 6= 0, such that kfk2 = 0.
To correct this, one identifies functions f and g if kf − gk2 = 0 and considers the quotient
vector space. We will not need to worry about this in the problems below.
P ROBLEM 1.
(1)Prove the Schwarz inequality: for any f and g in R(α),
hf, gi2 ≤ kfk2 kgk2 .
(Recall that for v, w ∈ Rn , v · w = kvkkwk cos θ where θ is the angle between the
vectors. The inequality states that | cos θ| ≤ 1.)
(2)Prove the triangle inequality: for any f, g and h in R(α),
kf − hk2 ≤ kf − gk2 + kg − hk2 .
P ROBLEM 2. Prove that continuous functions are dense with respect to the L2 metric: for
any f ∈ R(α) and ε > 0, there exists a continuous function g on [a, b], such that
kf − gk2 < ε.
(The hint in the textbook tells you to choose a partition P and take g piecewise linear, such
that g(xi ) = f(xi ) for all i = 0, . . . , n. To prove the inequality, it may be easier first to show
1
that for any η > 0 there exists g, defined by a suitable partition P, such that
Zb
|f − g|dα < η.
a
After that use the boundedness of f(x). This second inequality is easier to handle by
computing the upper and lower sums of the integral using the same partition P. The upper
sum can be compared to the difference between the upper and lower sums of f.)
Problems 3 and 4 explain how to define the Riemann-Stieltjes integral
necessarily monotonic.
Given a function g : [a, b] → R, define its variation on [a, b]
Vab (g) = sup
P
n
X
Rb
a
fdg when g is not
|g(xi ) − g(xi−1 )|,
i=1
where the supremum is over all partitions P = {x0 < · · · < xn } of [a, b]. We say that g has
bounded variation in [a, b] if Vab (g) is finite. Let BV be the set of all functions of bounded
variation on [a, b].
P ROBLEM 3. Prove the Jordan decomposition theorem: Any g ∈ BV can be decomposed as
g = α − β, where α and β are nondecreasing functions on [a, b]. You may use the functions
1
α(x) = Vax (g) + g(x) ,
2
1
β(x) = Vax (g) − g(x) ,
2
and prove that both α and β are non-decreasing. (Hint: The essential part in the proof is
the equality
Vab (g) = Vac (g) + Vcb (g)
for any a < c < b. We will prove this equality in class, so you can assume it.)
P ROBLEM 4. For g ∈ BV with decomposition g = α − β, and f continuous, define the
Riemann-Stieltjes integral
Zb
Zb
Zb
fdg = fdα − fdβ.
a
a
a
Prove that this does not depend on the choice of the decomposition: if g = α1 − β1 , with
α1 , β1 nondecreasing, then
Zb
Zb
Zb
Zb
fdα1 − fdβ1 = fdα − fdβ.
a
(Hint: use Riemann sums.)
a
a
a