Test 1: Take Home1. Let X denote the set of all irrational numbers x with √2 ≤ x ≤ 2√2, andwith the usual metric d(x, y) = |x − y|. Prove that X is not compact.2. Let (X, d) denote any metric space. The metric space X is called “totallybounded” when, for every � > 0, there exists finitely many neighborhoodsN�(xi) (i = 1, . . . n) such that X ⊆ ∪ni=1N�(xi). The metric space is“bounded” when { d(x, y) | x, y ∈ R } is a bounded subset of R.(a) Give an example of a bounded metric space that is not totally bounded.(b) Prove that every totally bounded metric space is bounded(c) Prove that a metric space is compact if and only if it is both completeand totally bounded.3. Let Rn denote the usual n-dimensional Euclidean space, with its Euclideannorm||x|| =vuutXni=1|xi|2an代写data留学生作业、代做R语言作业、代写R编程语言作业 代做留学生Processing|调试Matlab程序d corresponding metric d(x, y) = ||x − y||, with x, y ∈ Rn. Given ann × n matrix T, define||T|| ≡ sup { ||T x|| | ||x|| ≤ 1 } .(a) Prove that, for all n × n matrices X and Y , that ||XY || ≤ ||X||||Y||.(b) Prove that||T|| = inf { M ∈ R | ||T x|| ≤ M||x|| for all x ∈ Rn}.(c) With x ∈ Rn, find ||Cx|| when Cx is the n × n matrix with thecoordinates of x in the first column and zeros elsewhere.(d) With x ∈ Rn, find ||Dx|| when Dx is the n × n diagonal matrix withthe coordinates of x on the main diagonal, and zeros elsewhere.(e) With x ∈ Rn, find ||Rx|| when Rx is the n × n matrix with thecoordinates of x in the first row and zeros elsewhere.4. Let T be an n × n matrix, with ||T|| defined as in the previous problem.Prove that sup { |α| | α an eigenvalue of T }1转自：http://www.6daixie.com/contents/18/5015.html