Measure, Integration & Real Analysis, 2021a

Section 6E Consequences of Baire’s Theorem 189
6.85 Closed Graph Theorem
Suppose V and W are Banach spaces and T is a function from V to W. Then T is
a bounded linear map if and only if graph(T) is a closed subspace of V × W.
Proof First suppose T is a bounded linear map. Suppose ( f 1 , Tf 1 ), ( f 2 , Tf 2 ),...is
a sequence in graph(T) converging to ( f , g) ∈ V × W. Thus
lim f k = f and lim Tf k = g.
k→∞ k→∞
Because T is continuous, the first equation above implies that lim k→∞ Tf k = Tf;
when combined with the second equation above this implies that g = Tf. Thus
( f , g) =(f , Tf) ∈ graph(T), which implies that graph(T) is closed, completing
the proof in one direction.
To prove the other direction, now suppose graph(T) is a closed subspace of
V × W. Thus graph(T) is a Banach space with the norm that it inherits from V × W
[from 6.84 and 6.16(b)]. Consider the linear map S : graph(T) → V defined by
Then
S( f , Tf)= f .
‖S( f , Tf)‖ = ‖ f ‖≤max{‖ f ‖, ‖Tf‖} = ‖( f , Tf)‖
for all f ∈ V. Thus S is a bounded linear map from graph(T) onto V with ‖S‖ ≤1.
Clearly S is injective. Thus the Bounded Inverse Theorem (6.83) implies that S −1 is
bounded. Because S −1 : V → graph(T) satisfies the equation S −1 f =(f , Tf), we
have
‖Tf‖≤max{‖ f ‖, ‖Tf‖}
= ‖( f , Tf)‖
= ‖S −1 f ‖
≤‖S −1 ‖‖f ‖
for all f ∈ V. The inequality above implies that T is a bounded linear map with
‖T‖ ≤‖S −1 ‖, completing the proof.
Principle of Uniform Boundedness
The next result states that a family of
bounded linear maps on a Banach space
that is pointwise bounded is bounded in
norm (which means that it is uniformly
bounded as a collection of maps on the
unit ball). This result is sometimes called
the Banach–Steinhaus Theorem. Exercise
17 is also sometimes called the Banach–
Steinhaus Theorem.
The Principle of Uniform
Boundedness was proved in 1927 by
Banach and Hugo Steinhaus
(1887–1972). Steinhaus recruited
Banach to advanced mathematics
after overhearing him discuss
Lebesgue integration in a park.
Measure, Integration & Real Analysis, by Sheldon Axler