1. Theorem of Banach stainhaus and of Closed Graph
Chapter 2
Theorem of Banach stainhaus
and of Closed Graph

2. II.1Recall of Baire’s Lemma

3. Lemma II.1 (Baire) Let X be a complete metric space
and a seq. of closed sets.
Assume that for each n .
Then

6. Baire’s Category Theorem
Remark 1
Baire’s Category Theorem
Baire’s Lemma is usually used in
the following form. Let X be a
nonempty complete metric space
and a seq. of closed sets
such that Then there
is such that

7. II.2 The Theorem of Banach -Steinhaus

8. L(E,F) L(E,E)=L(E) Let E and F be two normed vector spaces.
Denoted by L(E,F) the space of all
linear continuous operators from
E to F equiped with norm
L(E,E)=L(E)

9. Theorem II.1(Banach Steinhaus)
Let E and F be two Banach space
and a family of linear
continuous operators from E to F
Suppose (1)
then (2)
In other words, there is c such that

12. Remark 2 In American literature, Theorem II.1
is referred as principle of uniform
boundness, which expresses well
the conceit of the result:
One deduces a uniform estimate
from pointwise estimates.

13. Corollary II.2 Let E and F be two Banach spaces and a family of linear
continuous operators from E to F
such that for each
converges as to a limit denoted
by Tx. Then we have

17. Corollary II.3 Let G be a Banach space and B a
subset of G. Suppose that
(3) For all f , the set
is bounded.(in R)
Then (4) B is bounded

19. Dual statement of corollary II.3
Let G be a Banach space and
a subset of Suppose that
(5) For all , the set
is bounded.
Then (6) is bounded.

21. II.3 Open Mapping Theorem
And Closed Graph Theorem

22. Theorem II.5 (Open Mapping Thm,Banach)
Let E and F be two Banach spaces
and T a surjective linear continuous
from E onto F. Then there is a
constant c>0 such that

27. Remark 4 Property (7) implies that T maps
each open set in E into open set in
F (hence the name of the Theorem)
In fact, let U be an open set in E,
let us prove that TU is open in T.

29. Corollary II.6 Let E and F be Banach spaces
be linear continuous and bijective. Then
is continuous from F to E

30. Argument by homogeneity

32. Remark 5 Let E be a vector space equiped with two norms and
Assume that and are
Banach space and assume that there
is C such that
Then there is c>0 such that

33. i.e.
and
are equivalent
Proof: Apply Corolary II.6 with
and T is identity

34. Graph G(T)
The graph G(T) of a linear
operator from E to F is the set

35. Theorem II.7 (Closed Graph Theorem)
Let E and F be two Banach spaces
and T a linear operator from E to F.
Suppose that the graph G(T) is
closed in
Then T is continuous.
(converse also holds)