# 数学代写：MATH5705 Mathematics and Statistics Assignment

## MATH5705 Mathematics and Statistics Assignment 代写案例

Questions：

1.Let X = ℓ 1 , the space of summable sequences x = (xk) ∞ k=1, with the norm

∥x∥1 = X∞ k=1 |xk|

Let Y = {x ∈ ℓ 1 : P∞ k=1 xk > 0}.

(a) Identify the interior points, the boundary points and the limit points of Y in X. Give proofs of your answers using the definitions of these concepts (Definitions 2.4.4 and 2.4.12).

(b) Is Y open? closed? bounded?

2. Consider the map f : (R, | · |) → (C[0, 2π], ∥·∥∞),

f(x)(t) = sin(x + t), x ∈ [0, 1], t ∈ [0, 2π].

Is f continuous at x = 1 2 ? Prove your answer.

3.Let p(z) = z 5 − 2z 4 − iz2 + 7z + 1 + i, z ∈ C. For r > 0 let

Dr = {z ∈ C : |z| ≤ r}.

(a) Let r0 = 0.19. Explain why p has no roots in Dr0 .

(b) Give as small a value of r as you can so that Dr contains exactly one root of p. (I don’t intend that you optimize r to 100 significant figures; being within 0.01 will do. It will suffice to numerically find the root z0 closest to the origin and check that r < |z0| + 0.01.)

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