Measure, Integration & Real Analysis, 2021a

Section 6E Consequences of Baire’s Theorem 191
5 Suppose
X = {0}∪
and d(x, y) =|x − y| for x, y ∈ X.
∞⋃ { 1k
}
(a) Show that (X, d) is a complete metric space.
(b) Each set of the form {x} for x ∈ X is a closed subset of R that has an
empty interior as a subset of R. Clearly X is a countable union of such sets.
Explain why this does not violate the statement of Baire’s Theorem that
a complete metric space is not the countable union of closed subsets with
empty interior.
6 Give an example of a metric space that is the countable union of closed subsets
with empty interior.
[This exercise shows that the completeness hypothesis in Baire’s Theorem cannot
be dropped.]
7 (a) Define f : R → R as follows:
⎧
⎪⎨ 0 if a is irrational,
f (a) = 1
⎪⎩
n
if a is rational and n is the smallest positive integer
such that a = m n
for some integer m.
At which numbers in R is f continuous?
(b) Show that there does not exist a countable collection of open subsets of R
whose intersection equals Q.
(c) Show that there does not exist a function f : R → R such that f is continuous
at each element of Q and discontinuous at each element of R \ Q.
8 Suppose (X, d) is a complete metric space and G 1 , G 2 ,... is a sequence of
dense open subsets of X. Prove that ⋂ ∞
k=1
G k is a dense subset of X.
9 Prove that there does not exist an infinite-dimensional Banach space with a
countable basis.
[This exercise implies, for example, that there is not a norm that makes the
vector space of polynomials with coefficients in F into a Banach space.]
10 Give an example of a Banach space V, a normed vector space W, a bounded
linear map T of V onto W, and an open subset G of V such that T(G) is not an
open subset of W.
[This exercise shows that the hypothesis in the Open Mapping Theorem that
W is a Banach space cannot be relaxed to the hypothesis that W is a normed
vector space.]
11 Show that there exists a normed vector space V, a Banach space W, a bounded
linear map T of V onto W, and an open subset G of V such that T(G) is not an
open subset of W.
[This exercise shows that the hypothesis in the Open Mapping Theorem that V
is a Banach space cannot be relaxed to the hypothesis that V is a normed vector
space.]
k=1
Measure, Integration & Real Analysis, by Sheldon Axler