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Theorem of Banach stainhaus and of Closed Graph
Chapter 2 Theorem of Banach stainhaus and of Closed Graph
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II.1Recall of Baire’s Lemma
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Lemma II.1 (Baire) Let X be a complete metric space
and a seq. of closed sets. Assume that for each n . Then
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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
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II.2 The Theorem of Banach -Steinhaus
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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)
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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
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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.
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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
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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
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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.
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II.3 Open Mapping Theorem
And Closed Graph Theorem
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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
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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.
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Corollary II.6 Let E and F be Banach spaces
be linear continuous and bijective. Then is continuous from F to E
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Argument by homogeneity
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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
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i.e. and are equivalent Proof: Apply Corolary II.6 with and T is identity
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Graph G(T) The graph G(T) of a linear operator from E to F is the set
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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)
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