数学final代考 Math 328 & Math 601代写 Math代写

Final Examination

Math 328 & Math 601

数学final代考 Directions: • Write your names and student number on the top right-hand corner of this page. • Open this booklet only when directed to do so.

Directions: 数学final代考

• Write your names and student number on the top right-hand corner of this page.

• Open this booklet only when directed to do so.

• Check that you have all 7 pages including this one.

• Write all your answers in the space provided.

• You may use the backs of sheets for rough work, or if you need additional space for your answer.

• Each question is worth 16 points. (Total: 100 points)

• No cheating sheets may be used.

• This examination is two days (48 hours) in duration.

4. 数学final代考

Determine whether the following statements true or false. Give reasons.

(a) Every bounded sequence in Rⁿ has a convergent subsequence.

(b) The dual space of (Rⁿ, ∥· ∥₁) is (Rⁿ, ∥· ∥₁).

(c) In a Hilbert space, every bounded sequence has a convergent subsequence.

5.

Suppose that the sequence (x_n)_n∈N in H satisfies

(a) There exists M > 0 such that for every n ∈ N, ∥ x_n∥ ≤ M.

(b) For every y ∈ M, lim_n→∞ 〈x_n, y〉exists.

(c) M is total in H.

Prove that for every y ∈ H, lim_n→∞ 〈x_n, y〉 exists.

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