Introduction to Analysis
Selected Homework Problems
1. Suppose X is a complete metric space, and {Gn} is a sequence of dense open subsets of X. Prove Baire’s theorem, namely, that ∩∞1 Gn is not empty. (In fact, it is dense in X.) 2. (Rudin ex6.10) Let p and q be positive real numbers such that
1 p+ 1
q = 1.
Prove the following statements.
(a) If u ≥ 0 and v ≥ 0, then
uv ≤ up p +vq
q . Equality holds if and only if up = vq.
(b) If f ∈ R(α), g ∈ R(α), f ≥ 0, g ≥ 0, and Z b
a
fpdα = 1 = Z b
a
gqdα, then
Z b a
f gdα ≤ 1.
(c) If f and g are real functions in R(α), then
Z b a
f gdα
≤nZ b a
|f |pdαo1/pnZ b a
|g|qdαo1/q
.
This is H¨older’s inequality. When p = q = 2 it is usually called the Schwarz inequal- ity.
3. (a) Suppose that f ∈ R(α) on [a, b], and that Rb
af (x)2dα(x) = 0. Prove that for all g ∈ R(α) one also hasRb
a f (x)g(x)dα(x) = 0.
(b) Deduce that if f ∈ R(α) on [a, b], and f (x) ≥ 0 for all x, then the following conditions are equivalent: (i) Rb
af (x)dα(x) = 0. (ii) Rb
af (x)2dα(x) = 0. (iii) Rb
a f (x)g(x)dα(x) = 0 for all g ∈ R(α).
(c) Show that if from part (b) we delete that assumption that f (x) ≥ 0 for all x, then the equivalence still holds, with f (x) replaced by |f (x)| in statement (i), but not in statements (ii) and (iii). (Hint: f = (|f | + f )/2) − (|f | − f )/2.)
4. (Rudin ex7.3)
(a) Suppose {fn} and {gn} are bounded sequences each of which converges uniformly on a set E in a metric space X to functions f and g, respectively, Show that the sequence {fngn} converges uniformly to f g on E.
(b) Construct sequences {fn}, {gn} which converge uniformly on some set E, but such that {fngn} does not converge uniformly on E (of course, {fngn} must converge on E).
5. (Rudin ex7.9) Let {fn} be a sequence of continuous functions which converges uniformly to a function f on a set E. Prove that
n→∞lim fn(xn) = f (x)
for every sequence of points xn∈ E such that xn→ x, and x ∈ E. Is the converse of this true?
6. (Rudin ex7.12) Suppose g and fn(n = 1, 2, 3, · · · ) are defined on (0, ∞), are Riemann- integrable on [t, T ] whenever 0 < t < T < ∞, |fn| ≤ g, fn → f uniformly on every compact subset of (0, ∞), and
Z ∞ 0
g(x)dx < ∞.
Prove that
n→∞lim Z ∞
0
fn(x)dx = Z ∞
0
f (x)dx.
7. (Rudin ex7.13) Assume that {fn} is a sequence of monotonically increasing functions on R1 with 0 ≤ fn(x) ≤ 1 for all x and all n.
(a) Prove that there is a function f and a sequence {nk} such that f (x) = lim
k→∞fnk(x) for every x ∈ R1.
(b) If, moreover, f is continuous, prove that fnk → f uniformly on R1.
8. (Rudin ex7.19) Let K be a compact metric space, let S be a subset of C(K). Prove that S is compact if and only if S is uniformly closed, pointwise bounded, and equicontinuous.
(If S is not equicontinuous, then S contains a sequence which has no equicontinuous subsequence, hence has no subsequence that converges uniformly on K.)
9. Let X be a metric space and suppose that f : X → R1 has the following property: there are positive numbers M , α with 0 < α ≤ 1 such that
|f (p) − f (q)| ≤ M [d(p, a)]α
for any two points p, q in X. Then we say that f satisfies a H¨older condition on X with constant M and exponent α. Show that a family F of functions satisfying a H¨older condition with constant M and exponent α is equicontinuous.
10. Suppose f is a continuous complex-valued function on the real line. Show that there exists a sequence of polynomials Pn such that for each finite interval [a, b], the polynomials Pn converge uniformly to f on [a, b].
11. Let f(x, y) = (2x2 + 10y + xy, x3 + y3 + 20x). Prove that f 0(x, y) is invertible for all
−1 ≤ x, y ≤ 1.
12. Let L : Rn → Rn be a linear isomorphism and f (x) = L(x) + g(x), where |g(x)| ≤ M |x|2 and f is C1. Show that f is invertible near 0. (We say that h : Rn→ Rnis an isomorphism if |h(x) − h(y)| = |x − y|.)
13. Decide whether the system
u(x, y, z) = x + xyz, v(x, y, z) = y + xy, w(x, y, z) = z + 2x + 3z2 can be solved for x, y, z in terms of u, v, w near (0, 0, 0).
14. Let A be a compact subset in Rn and let B be a compact subset in C(A). Show that there are an f0 ∈ B and an x0 ∈ A such that g(x) ≤ f0(x0) for all g ∈ B and x ∈ A.
15. (a) In R2, consider the set B = {(x1, x2)
x21+ x22 ≤ 1}. Show that the function f (x) = αx21+ βx22, α > 0, β > 0 defined in B is a convex function.
(b) In R2, consider the square S = [0, 1] × [0, 1]. Show that the function g(x) = x1x2 is not a convex function in S.
