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Elementary Geometry For College Students, 7e
7th Edition
ISBN: 9781337614085
Author: Alexander, Daniel C.; Koeberlein, Geralyn M.
Publisher: Cengage,
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
Transcribed Image Text: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.
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