Measure, Integration & Real Analysis, 2021a

186 Chapter 6 Banach Spaces
Because
R = ⋃
{x}
x∈R
and each set {x} has empty interior in R, Baire’s Theorem implies R is uncountable.
Thus we have yet another proof that R is uncountable, different than Cantor’s original
diagonal proof and different from the proof via measure theory (see 2.17).
The next result is another nice consequence of Baire’s Theorem.
6.80 the set of irrational numbers is not a countable union of closed sets
There does not exist a countable collection of closed subsets of R whose union
equals R \ Q.
Proof This will be a proof by contradiction. Suppose F 1 , F 2 ,... is a countable
collection of closed subsets of R whose union equals R \ Q. Thus each F k contains
no rational numbers, which implies that each F k has empty interior. Now
( ⋃ ) ( ⋃ ∞
R = {r} ∪ F k
).
r∈Q
k=1
The equation above writes the complete metric space R as a countable union of
closed sets with empty interior, which contradicts Baire’s Theorem [6.76(a)]. This
contradiction completes the proof.
Open Mapping Theorem and Inverse Mapping Theorem
The next result shows that a surjective bounded linear map from one Banach space
onto another Banach space maps open sets to open sets. As shown in Exercises 10
and 11, this result can fail if the hypothesis that both spaces are Banach spaces is
weakened to allow either of the spaces to be a normed vector space.
6.81 Open Mapping Theorem
Suppose V and W are Banach spaces and T is a bounded linear map of V onto W.
Then T(G) is an open subset of W for every open subset G of V.
Proof Let B denote the open unit ball B(0, 1) ={ f ∈ V : ‖ f ‖ < 1} of V. For any
open ball B( f , a) in V, the linearity of T implies that
T ( B( f , a) ) = Tf + aT(B).
Suppose G is an open subset of V. Iff ∈ G, then there exists a > 0 such that
B( f , a) ⊂ G. If we can show that 0 ∈ int T(B), then the equation above shows that
Tf ∈ int T ( B( f , a) ) . This would imply that T(G) is an open subset of W. Thus to
complete the proof we need only show that T(B) contains some open ball centered
at 0.
Measure, Integration & Real Analysis, by Sheldon Axler