Spectral Theory in Hilbert Space

EXERCISES FOR APPENDIX B 127
Exercises for Appendix B
Exercise B.1. Prove Proposition B.1.
Exercise B.2. Prove the calculation rules (1)–(5).
Exercise B.3. Prove Proposition B.2.
Exercise B.4. Show that if f and g has a common point of discontinuity
in [a, b], then f is not Riemann-Stieltjes integrable with respect
to g over [a, b].
Exercise B.5. Show that if f is absolutely continuous on [a, b],
then f is of bounded variation on [a, b], and V b
a (f) = b
a |f ′ |.
Hint: First show V b
a (f) ≥ b
a |f ′ |. To show the other direction, write
(B.1) on the form b
a ϕf ′ for a stepfunction ϕ and use Hölder’s inequality.
Exercise B.6. Show that the set of all functions of bounded variation
on an interval [a, b] is made into a normed linear space by setting
f = |f(a)| + V b
a (f). Convergence in this norm is called convergence
in variation. Show that convergence in variation implies uniform convergence,
and that the normed space just introduced is complete (any
Cauchy sequence of functions in the space converges in variation to a
function of bounded variation).
Exercise B.7. Show that a monotone function can have at most
countably many discontinuities, all of them jump discontinuities. Also
show that if a function of bounded variation is continuous to the left
(right) at a point, then so are its positive and negative variation functions,
and that only if the function jumps up (down) will the positive
(negative) variation function have a jump.
Hint: How many jumps of size > 1/j can there be?
Exercise B.8. Prove Theorem B.8. Also show that if g is absolutely
continuous on [a, b], then any Riemann integrable f is integrable
with respect to g and the same formula holds.
Hint: f(ξk)(g(xk) − g(xk−1) = ϕg ′ where ϕ is a step function
converging to f.
Exercise B.9. Suppose f, g are continuous and ρ of bounded variation
in (a, b). Put σ(t) = t
f(s) dρ(s) for some c ∈ (a, b). Show that
c
b
a
g(t) dσ(t) =
b
a
g(t)f(t) dρ(t) .
Hint: Integrate both sides by parts, first replacing (a, b) by an arbitrary
compact subinterval.