13-dual-ban

13-dual-ban - Lecture 13 4 The weak dual topology In this...

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Unformatted text preview: Lecture 13 4. The weak dual topology In this section we examine the topological duals of normed vector spaces. Be- sides the norm topology, there is another natural topology which is constructed as follows. Definition. Let X be a normed vector space over K (= R , C ). For every x ∈ X , let x : X * → K be the linear map defined by x ( φ ) = φ ( x ) , ∀ φ ∈ X * . We equipp the vector space X * with the weak topology defined by the family Ξ = ( x ) x ∈ X . This topology is called the weak dual topology , which is denoted by w * . Recall (see Section 3) that this topology is characterized by the following property ( w * ) Given a topological space T , a map f : T → X * is continuous with respect to the w * topology, if and only if x ◦ f : T → K is continuous, for each x ∈ X . Remark that all the maps x : X * → K , x ∈ X are already continuous with respect to the norm topology. This gives the fact that • the w * topology on X * is weaker than the norm topology. Remark 4.1. The w * topology is Hausdorff. Indeed, if φ,ψ ∈ X * are such that φ 6 = ψ , then there exists some x ∈ X such that x ( φ ) = φ ( x ) 6 = ψ ( x ) = x ( ψ ) . Proposition 4.1. Let X be a normed vector space over K . For every ε > , φ ∈ X * , and x ∈ X , define the set W ( φ ; x,ε ) = ψ ∈ X * : | ψ ( x )- φ ( x ) | < ε. Then the collection W = W ( φ ; x,ε ) : ε > , φ ∈ X * , x ∈ X is a subbase for the w * topology. More precisely, given φ ∈ X * , a set N ⊂ X * is a neighborhood of φ with respect to the w * topology, if and only if, there exist ε > and x 1 ,...,x n ∈ X , such that N ⊃ W ( φ ; ε,x 1 ) ∩ ··· ∩ W ( φ ; ε,x n ) . Proof. It is clearly sufficient to prove the second assertion, because it would imply the fact that any w * open set is a union of finite intersections of sets in W . If we define the collection S =- 1 x ( D ) : x ∈ X , D ⊂ K open , then we know that S is a subbase for the w * topology. 89 90 LECTURE 13 Fix φ ∈ X * . Start with some w * neighborhood N of φ , so there exists some w * open set E with φ ∈ E ⊂ N . Using the fact that S is a subbase for the w * topology, there exist open sets D 1 ,...,D n ⊂ K , and points x 1 ,...,x n , such that φ ∈ n k =1- 1 x k ( D k ) ⊂ E. Fix for the moment k ∈ { 1 ,...,n } . The fact that φ ∈- 1 x k ( D k ) means that φ ( x k ) ∈ D k . Since D k is open in K , there exists some ε k > 0, such that D k ⊃ B ε k ( φ ( x k ) ) . Then if we have an arbitrary ψ ∈ W ( φ ; ε k ,x k ), we will have | ψ ( x k )- φ ( x k ) | < ε k , which gives ψ ∈- 1 x k ( D k ). This proves that W ( φ ; ε k ,x k ) ⊂- 1 x k ( D k ) . Notice that, if one takes ε = min { ε 1 ,...,ε n } , then we clearly have the inclusions W ( φ ; ε,x k ) ⊂ W ( φ ; ε k ,x k ) ⊂- 1 x k ( D k ) ....
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This note was uploaded on 10/11/2010 for the course MATH 11 taught by Professor Smith during the Three '10 term at ADFA.

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13-dual-ban - Lecture 13 4 The weak dual topology In this...

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