We denote by 0 the set of all sequences u = (ux)k=1,2, = (u1, u2, ...) (uk E R) %3D satisfying sup Ju< 00. Moreover, we define a norm of as follows: I| = sup ul. Then, prove that is a Banach space with respect to || - ||- (It is sufficient to prove com- pleteness of , i.e., show that a Cauchy sequence {u}-1.2 is a convergent sequence in , where un) = (u", u",..) E e*.) %3D

We denote by 0 the set of all sequences u = (ux)k=1,2, = (u1, u2, ...) (uk E R) %3D satisfying sup Ju< 00. Moreover, we define a norm of as follows: I| = sup ul. Then, prove that is a Banach space with respect to || - ||- (It is sufficient to prove com- pleteness of , i.e., show that a Cauchy sequence {u}-1.2 is a convergent sequence in , where un) = (u", u",..) E e*.) %3D

Elements Of Modern Algebra

8th Edition

ISBN:9781285463230

Author:Gilbert, Linda, Jimmie

Publisher:Gilbert, Linda, Jimmie

Chapter2: The Integers

Section2.3: Divisibility

Problem 33E

See similar textbooks

Similar questions

Let R be the set of all infinite sequences of real numbers, with the operations u+v=(u1,u2,u3,......)+(v1,v2,v3,......)=(u1+v1,u2+v2,u3+v3,.....) and cu=c(u1,u2,u3,......)=(cu1,cu2,cu3,......). Determine whether R is a vector space. If it is, verify each vector space axiom; if it is not, state all vector space axioms that fail.
Proof Prove that if S1 is a nonempty subset of the finite set S2, and S1 is linearly dependent, then so is S2.
6. In Example 3 of section 3.1, find elements and of such that but . From Example 3 of section 3.1: and is a set of bijective functions defined on .
a)To prove this directly, you'd suppose that A is a subset of B. Then you'd need to show that sup(A)≤sup(B)sup(A)≤sup(B). Since A is bounded above, sup(A) is the least upper bound for A which, by definition of l.u.b., is less than or equal to all upper bounds for A. b)Note that, max{sup(A), sup(B)} is either sup(A) or sup(B). So you have a couple of natural cases to break things down into. c) I am not sure of
This 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.)
Let (X,T) be a topological space, K a compact subset of X and (Fn)n∈ N a family in P(X) with Fn≠ø for all n. If K⊇F̅0⊇F̅1⊇....⊇F̅n⊇........ then prove that (by way of contradiction) n=0∩∞ F̅n ≠ø
16. The set S = { x∈R: x2 - 4<0} with the usual metric is .......................... A. Compact. B. Connected. C. Not connected. D. Sequentially compact.
Consider the space Z+ with the finite complement topology. Consider the sequence (xn) of points in Z+ given by xn = n+7. To what point or points does the sequence converge?
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.
Prove that if A ⊆ B and both sets have minima then min B ≤ min A
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.
Let T and T 'be two topologies of a set X.Is the family T U T´formed by the openings common to both also a topology of X?