Measure, Integration & Real Analysis, 2021a

188 Chapter 6 Banach Spaces
The next result provides the useful information
that if a bounded linear map
The Open Mapping Theorem was
first proved by Banach and his
from one Banach space to another Banach
colleague Juliusz Schauder
space has an algebraic inverse (meaning
(1899–1943) in 1929–1930.
that the linear map is injective and surjective),
then the inverse mapping is automatically bounded.
6.83 Bounded Inverse Theorem
Suppose V and W are Banach spaces and T is a one-to-one bounded linear map
from V onto W. Then T −1 is a bounded linear map from W onto V.
Proof The verification that T −1 is a linear map from W to V is left to the reader.
To prove that T −1 is bounded, suppose G is an open subset of V. Then
(T −1 ) −1 (G) =T(G).
By the Open Mapping Theorem (6.81), T(G) is an open subset of W. Thus the
equation above shows that the inverse image under the function T −1 of every open
set is open. By the equivalence of parts (a) and (c) of 6.11, this implies that T −1 is
continuous. Thus T −1 is a bounded linear map (by 6.48).
The result above shows that completeness for normed vector spaces sometimes
plays a role analogous to compactness for metric spaces (think of the theorem stating
that a continuous one-to-one function from a compact metric space onto another
compact metric space has an inverse that is also continuous).
Closed Graph Theorem
Suppose V and W are normed vector spaces. Then V × W is a vector space with
the natural operations of addition and scalar multiplication as defined in Exercise 10
in Section 6B. There are several natural norms on V × W that make V × W into a
normed vector space; the choice used in the next result seems to be the easiest. The
proof of the next result is left to the reader as an exercise.
6.84 product of Banach spaces
Suppose V and W are Banach spaces. Then V × W is a Banach space if given
the norm defined by
‖( f , g)‖ = max{‖ f ‖, ‖g‖}
for f ∈ V and g ∈ W. With this norm, a sequence ( f 1 , g 1 ), ( f 2 , g 2 ),... in
V × W converges to ( f , g) if and only if lim f k = f and lim g k = g.
k→∞ k→∞
The next result gives a terrific way to show that a linear map between Banach
spaces is bounded. The proof is remarkably clean because the hard work has been
done in the proof of the Open Mapping Theorem (which was used to prove the
Bounded Inverse Theorem).
Measure, Integration & Real Analysis, by Sheldon Axler