泛函分析代写 Functional Analysis代写 Class Test 代写

MT4515 Functional Analysis – Class Test 1

泛函分析代写 Answer all questions. There are 3 questions, each carrying 5 marks. You should justify your answers which should be presented clearly and logically.

Answer all questions. There are 3 questions, each carrying 5 marks. You should justify your answers which should be presented clearly and logically.

Throughout these questions the function spaces considered should be taken to be of realvalued functions.

Q1. 泛函分析代写

Let C[−1, 1] denote the vector space of real-valued continuous functions on the interval [−1, 1]. For f ∈ C[−1, 1] define

Show that ‖ ‖ is a norm on C[−1, 1].

Do you think C[−1, 1] is complete with this norm? [Here only a very brief reason, not a full proof, is required, e.g. by analogy with another space that you may have met.]

Q2. 泛函分析代写

Express the solutions of the simultaneous equations

6x − 2 cos x − y² = 0

6y − 2 sin y − x² = 0

as fixed points of a certain mapping T : R² → R² . Show that these equations have exactly one solution (x, y) for which 0 ≤ x, y ≤ 1 (ignore other solutions outside this range).

[Recall that the contraction mapping theorem is valid on the complete set [0, 1] × [0, 1], ‖ ‖_∞
, with [0, 1] × [0, 1] the closed unit square in R² and the norm ‖(x, y)‖_∞ = max{|x|, |y|}.]

[You may assume that the Stone-Weierstrass theorem is valid on C([0, 1]×[0, 1]), ‖ ‖_∞
where ‖f‖_∞ = sup_0≤x,y≤1 |f(x, y)|.]

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