Problem 1: Connectedness and Compactness in Topological Spaces Consider the topological space X = R² with the standard topology. Let A and B be subsets of X defined as follows: • • A is the open unit disk, i.e., A = {(x, y) = R² | x² + y² < 1}. B is the closed annulus, i.e., B = {(x, y) Є R² | 1 ≤ x² + y² ≤ 4}. 1. Prove that A is connected but not compact. 2. Prove that B is compact. 3. Let C AUB. Determine whether C is connected and justify your reasoning. 4. Draw the sets A, B, and C, and illustrate their topological properties using a Venn diagram or a similar graphical representation. Problem 2: Banach Spaces and Weak Convergence Let X be a Banach space, and let {n} be a sequence in X such that it converges weakly to some x = X, i.e., xnx. Assume that {n} is also bounded, meaning sup ||xn|| < ∞. 1. Prove that weak convergence does not necessarily imply strong convergence in a Banach space, by providing a specific example in the space IP (with 1 < p < ∞). 2. Show that if X is a Hilbert space, then a sequence {n} converges weakly to a if and only if (xn, y) (x, y) for all y € X, where (.,.) denotes the inner product. 3. Illustrate the distinction between weak and strong convergence with a graph showing the convergence of norms versus inner product convergence for different sequences.