3. In the notation of Problem 2, suppose that V is a Banach space with respect to the norm || |v and W is a Banach space with respect to the norm || ||w. Show that V x W is a Banach space with respect to the norm || ||.
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- 16. The set S = { x∈R: x2 - 4<0} with the usual metric is .......................... A. Compact. B. Connected. C. Not connected. D. Sequentially compact.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≠0Suppose X is a topological space, Y is a Hausdorff space, and f:X→Y is continuous. Show that the graph {(x,f(x))∣x∈X} is closed.
- Show R³ is a banach space.Related to the solution for exercise problem 12, in section 7.4 of textbook "Discrete Mathematics: Introduction to Mathematical Reasoning, 4th Edition": What is the process for how the function "f(x) = (b - a)x + a" was assumed given the following information: S denotes the set of real numbers strictly between 0 and 1. That is, S = {x ∈ R | 0 < x < 1}. Let a and b be real numbers with a < b, and suppose thatW = {x ∈ R| a < x < b}. Prove that S and W have the same cardinality. I understood the later steps of proving the function being one-to-one and onto, but not sure how the function f(x) came to be in the first place.Show that R^3 is a banach space with explanation.
- Show that ℓ^1 is a normed linear space.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.Problem 9For the following equivalence relation describe the corresponding partition without anyredundancy or reference to the name of the relation. Let ∼be the relation on R−{0}definedby x ∼y if and only if xy > 0, for all x, y ∈R−{0}.