Real Analysis Qualifying Exam; August, 2009.
Work as many of these ten problems as you can in four hours. Start each problem on a new
sheet of paper.
#1.Evaluate the iterated integral
Z∞
0Z∞
0
xexp(−x2(1 + y2)) dx dy.
(Justify your answer.)
#2.Let f∈C[0,1] be real–valued. Prove that there is a monotone increasing sequence of
polynomials {pn(x)}∞
n=1 converging uniformly on [0,1] to f(x).
#3.Let {fn}∞
n=1 be a sequence of non-zero elements of L2[0,1]. Prove that there is a function
g∈L2[0,1] such that for all n≥1 we have
Z1
0
g(x)fn(x)dx 6= 0.
#4.Let (X, Σ, µ) be a measure space with µ(X)<∞. Given sets Ai∈Σ, i≥1, prove that
µ
∞
\
i=1
Ai= lim
n→∞
µ
n
\
i=1
Ai.
Give an example to show that this need not hold when µ(X) = ∞.
#5.Let Kbe a compact subset of Rnand describe the dual space of the Banach space
C(K). (You may choose either the real or the complex Banach space.)
Let 1∈C(K) denote the constant function taking value 1 and let Sbe the subset of the
dual space consisting of the positive bounded linear functionals on C(K) that map 1to 1.
Show that the extreme points of Sare the point evaluation maps, f7→ f(x).
#6.Let ℓ2(Z) denote the real Hilbert space of square–summable functions on the integers.
Let xk(k≥1) be a sequence in ℓ2(Z) that converges coordinate–wise to zero, i.e., such that
limk→∞ xk(n) = 0 for all n∈Z.
Must xkconverge in norm to 0 as k→ ∞? What about if kxkkis assumed to be bounded?
Must xkconverge weakly to 0 as k→ ∞? What about if kxkkis assumed to be bounded?
Justify your answers (by proof or counter-example.)
#7.Let Xbe a second countable (that is, having a countable basis of open sets) and normal
topological space. Show that there is a countable family Fof countinuous functions from X
into the interval [0,1] that separates points and closed sets: i.e., such that if x∈Xand C
is a closed subset of Xwith x6∈ C, then there is f∈ F such that f(x) = 0 and f(C)⊆ {1}.
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