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- 1. Suppose E⊆X , where X is a metric space, p is a limit point of E , f and g are complex functions on E and fx=A and gx=B . Prove fgx=AB if B≠0This is a real analysis question. Let (X,d) be a complete metric space with X not ∅. Suppose the function f : X → X has the property that there exists a constant C ∈ (0, 1) such that d(f(x), f(y)) ≤ Cd(x, y) for all x,y ∈ X. The goal of this problem is to prove that there exists a unique x^∗ ∈ X (a) Let x0 ∈ X be arbitrary. If the sequence {xn, n ∈ N} is defined by setting xn = f(xn−1) for n ∈ N, prove that {xn, n ∈ N} is Cauchy. (b) Since the metric space (X, d) is assumed to be complete, define the limit of the sequence {xn, n ∈ N} from (c) to be x^∗. Prove that f(x^∗) = x^∗. (This establishes existence.)16. The set S = { x∈R: x2 - 4<0} with the usual metric is .......................... A. Compact. B. Connected. C. Not connected. D. Sequentially compact.
- Let (X, B, μ) be a measure space. Prove the following: If{Ei} is a countable collection of sets in B with μ(E1) < ∞and Ei ⊇ Ei+1 for i = 1, 2,..., thenμ ∞i=1Ei= limn→∞ μ(En).Let (X,d) be a metric space. For x,y ∈ X, let D(x, y) = √d(x, y). Prove that D is a metric on X. Explain why, for a sequence (xn), n∈N in X, (xn) converges in the metric space (X,d) if and only if it converges in the metric space (X, D).Suppose (S,d) is a metric space. How can we prove that S is open?
- Let (X,d) be a metric space , x ϵ X and A ⊑ X be a nonempy set. Prove that d (x ,A) = 0 if and only if every neighborhood of x contains a point of A.Suppose, as a rough estimate, we say that there are 20 distinct geons used for object recognition; and a pair of geons can be placed in 10 distinct qualitative relations (geon A on top of geon; geon A to upper left of geon B; geon A to the left of geon B; and so forth). how many distinct two-geon objects do we have in the space described above? R = 100000 Now, suppose we add a third geon, geon C. Again, each geon comes in 20 varieties and 5 sizes. we'll start by creating a tow-geon pair of A and B just like in Question 1 above; then, we decide which of A or B the third geon (C) will be adjacent to, and then we place geon C beside either A or B in one of the 10 allowed relatios. how many distinct three-geon objects do we have in this space? R = ?Suppose E is a subset of X, where X is a metric space, p is a limit point of E, f and g are complex functions on E and the limit as x approaches p of f(x) is A and the limit as x apporaches p of g(x) is B. Prove the limit as x approaches p of (f/g)(x)=A/B if B does not equal 0.
- d((x1, x2, x3), (y1, y2, y3) = |x1 - y1| + |x2 - y2| + |x3 - y3|. Conclude that, with this metric, a subset of ℝ3 is sequentially compact if and only if it is closed and bounded.In formally proving that lim x→1 (x2 + x) = 2, let ε > 0 be arbitrary. Choose δ = min (ε/m, 1).Determine the smallest value of m that would satisfy the proof.We know that the set S = {1/n : n ∈ N} is not compact because 0 is a limit point of S that is not in S. To see the non-compactness of S in another way, find an open cover of S that does not have a finite subcover!
