ANALYSIS QUALIFYING EXAM PROBLEMS BRIAN LEARY ...

ANALYSIS QUALS 25
b) I present the proof in the (simple) case in which H is a separable
Hilbert space (which is how Stein-Shakarchi defines their Hilbert spaces).
In this case, let {en} be an orthonormal basis for H.
Define Tn(y) = 1
y 〈en, y〉 y. Then
Tn(y) ≤ y , and Tn(en) = 1, so Tn = 1.
But by Bessel’s inequality,
n
|〈en, y〉| 2 ≤ y 2 , so 〈en, y〉 → 0 for all y.
Thus, lim Tn(y) = 0 for all y ∈ H.
Problem 3: Let X be a Banach space and let X ∗ be its dual Banach space.
Prove that if X ∗ is separable the X is separable.
Solution. Let {fn} be a countable dense set of X ∗ . For each n, choose
xn ∈ X such that xn = 1 and |fn(xn)| ≥ 1
2 fn.
Let S be the set of all linear combinations of the xn with rational coefficients
(rational real and imaginary parts if X is a Banach space over C).
Thus, S is countable. We aim to show that S is dense in X.
Let Y = S and assume for sake of contradiction that Y = X. Then by
Hahn-Banach theorem, we may choose f ∈ X∗ such that f(y) = 0 for all
y ∈ Y and f = 0. Then by density of {fn}, choose a sequence fin → f.
Then
1
f − fin ≥ |(f − fin )(xin )| = |fin (xin )| ≥ fin 2
for all n ∈ N. But then as fin → f, we must have fin → 0, which means
that f = 0, a contradiction. Thus Y = X.
Problem 4: Let f(x) be a non-decreasing function on [0, 1]. You may assume
the theorem that f is differentiable almost everywhere.
a) Prove that 1
0 f ′ (x)dx ≤ f(1) − f(0).
b) Let {fn} be a sequence of non-decreasing functions on the unit interval
[0, 1], such that the series F (x) = ∞
n=1 fn(x) converges for all x ∈
[a, b]. Prove that F ′ (x) = ∞
n=1 f ′ n(x) almost everywhere on [0, 1].
Proof. a) Define the difference quotient
gn(x) =
f(x + 1/n) − f(x)
,
1/n
and then by the theorem we are permitted to assume, gn(x) → f ′ (x)
as n → ∞ for a.e. x. As f is non-decreasing, we have that gn is
non-negative, so by Fatou, we have that
1
0
f ′ (x)dx ≤ lim inf
1
0
gn(x)dx.