07-08学年第2学期《泛函分析(双语)》A卷[1]1

苏州大学泛函分析(双语)课程试卷(A)卷共6页

(考试形式闭卷

2008年7月)

院系年级专业学号

姓名

成绩

Note:Do 7problems totally,omit 1from Problems 7and 8.State which problem you omit.

1.(15marks)Suppose that E is a nonempty compact set in a metric space (X,d ).Prove that E is bounded in X .Furthermore,show that there exist x,y ∈E such that d (x,y )=δ(E ),where δ(E )is the diameter of E de?ned by δ(E )=sup u,v ∈E

d (u,v ).

1/6

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2.(15marks)Suppose that (X, · )is a Banach space,T :X →X .Prove that if

α0=inf n sup x =y T n x ?T n y

x ?y <1,

then T has a unique ?xed point on X ,i.e.T x =x has a unique solution

x ∈X .

3.(10marks)Show that the standard norm on C [a,b ]is not induced by an inner product.

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4.(15marks)Let H be a Hilbert space and let {e n }n ∈N be an orthonormal

sequence in H .Then {e n :n ∈N }⊥={0}if and only if x 2

=∞ n =1

| x,e n |2

for all x ∈H .

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5.(20marks)Let c 0be the space of sequences of numbers convergent to zero with the usual supremum norm x =sup {|x n |:n ∈N }(x ={x n }∈c 0),and let f :c 0→R de de?ned by

f (x )=∞ n =1

x n

2n ,?x ={x n }∈c 0.

i)Show that f is a continuous linear functional.ii)Find the norm of f .

iii)Show that f does not attain its norm on the closed unit ball.

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6.(10marks)Let H be a Hilbert space.Suppose that x 0,x n ∈H (n ∈N ).Prove that if {x n }converges weakly to x 0(n →∞),then {x n }converges strongly to x 0(n →∞)if and only if x n → x 0 (n →∞).

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7.(Option!15marks)Let X and Y be Banach spaces and let T :X →Y be bijective.Show that there exist positive constants a ,b such that ?x ∈X ,

a x T x

b x .

8.(Option!15marks)Let H be a Hilbert space and let T ∈B (H )be a self-adjoint operator.Show that all eigenvalues of T are real and eigenvectors corresponding to di?erent eigenvalues are orthogonal.

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