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- a. Is there any relation between reflexive normed space and a Banach Space? (If yes then prove) b. Give two examples of normed spaces that are not reflexive (with brief reasoning).Let (R,d) be diserete metric space then R is not compact . True or false.??Consider the collection of open sets, (-n,n) from n=1 to infinity in R with the usual metric d(x,y) = abs(x-y). Use this collection of open sets to show R is not compact (make sure to prove R = UneJ(-n,n) as part of your work.
- Let (X,T) be a topological space Property C=P.C. A subset A of x has P.C If it's subset of the union of two disjoint nonempty open subsets of X then A is contained in only one of these open sets. Prove the following; If A and B have P.C and A̅̅∩B≠ø then A∪B has property c.(a) Supply a definition for bounded subsets of a metric space (X, d). (b) Show that if K is a compact subset of the metric space (X, d), then K is closed and bounded. (c) Show that Y ⊆ C[0, 1] from Exercise 8.2.9 (a) is closed and bounded but not compact.Let ?1and ?2 be compact subsets of a metric space X. Show ?1 ∪ ?2 is compact
- Consider the setsS1 = {(x, y) : x^2 + y^2 < 1} andS2 = {(x, y) : y ≥ −x − 2, x ≤ 0, y ≤ 0}(a) Sketch each of the sets. With references to your drawings and open ε-balls, conclude whether they are open or closed. For any closed set, motivate whether it is compact ornot. (b) Is clS1 (intersect) clSc1 open or closed? What is this set called? (c) Is intS2US1 open or closed? Motivate.This exercise demonstrates the concepts of boundary point, open and closed sets, etc., highly dependent on X's mother space. Give a reason for its correctness.Suppose Y=[ 0 ,2 ) . In this case A=[1,2 ) is not closed in X; while closed in Y. In addition, G=[0 ,1 )is not open in X while it is open in Y.Suppose X=R^2 and Y=R . In this case A=(0 ,1) is not open in X while it is open in Y. In fact, inside A in X is empty!Do the following task.a) State formally the Heine-Borel theorem in R^kb) Apply Heine-Borel theorem to show that if A is bounded, B isclosed such that B ⊆ A ⊆ R^k, then B is compact.c) Give an example of set A and B in R such that A is bounded,B is closed, B ⊆ A but A is not compact.
- Let R` be the set of real numbers with the lower limit topology(the collection of the intervals [a, b) is the basis).(a) Show that the interval (1, 6) is open in R`. (b) What can you say about the interval (1, 6]? Justify your answer. (c) What about the interval (−∞, 3)?Show Corollary 1.24. Namely, show that a measure space (X, M , μ) is complete if (X, M , μ) is constructed via Carath ́eodory’s theo- rem (Theorem 1.20). Corollary 1.24. Let (X, M , μ) be a measure space obtained via Theo- rem 1.20. Then (X, M , μ) is complete. Theorem 1.20 (Carath ́eodory’s theorem). Let M be as above. We have (1) M is a σ-algebra.(2) ForE∈M,defineμ(E):=ν(E). ThenμisameasureonM.State whether each of the following statements is true or false, in either case substan-tiate/ justify.(g) The empty set is path-connected. (h) The set of natural numbers as a subset of real numbers with the usual metricis compact.(i) Let f and g be uniformly continuous on a metric space X into R . Then theproduct f ∗ g is uniformly continuous on X into R.