Let X be a Banach space and TE B(X) an operator with ||T|| E o(T). Show that || id +T|| = 1+ ||T||.
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A: Note: Since we only answer up to 3 sub-parts, we'll answer the first 3. Please resubmit the question…
Q: 3. Let X be a Banach space and TE B(X) an operator with ||T|| (T). Show that || id +T|| = 1+ ||T||.
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- Let f:AA, where A is nonempty. Prove that f a has right inverse if and only if f(f1(T))=T for every subset T of A.Complete the proof of Theorem 5.30 by providing the following statements, where and are arbitrary elements of and ordered integral domain. If and, then. One and only one of the following statements is true: . Theorem 5.30 Properties of Suppose that is an ordered integral domain. The relation has the following properties, whereand are arbitrary elements of. If then. If and then. If and then. One and only one of the following statements is true: .For each of the following parts, give an example of a mapping from E to E that satisfies the given conditions. a. one-to-one and onto b. one-to-one and not onto c. onto and not one-to-one d. not one-to-one and not onto
- 27. Let , where and are nonempty. Prove that has the property that for every subset of if and only if is one-to-one. (Compare with Exercise 15 b.). 15. b. For the mapping , show that if , then .In Example 3 of Section 3.1, find all elements a of S(A) such that a2=e. From Example 3 of section 3.1: A=1,2,3 and S(A) is a set of bijective functions defined on A.Prove statement d of Theorem 3.9: If G is abelian, (xy)n=xnyn for all integers n.
- 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≠0Let g be defined on an interval A, and let c ∈ A. (a) Explain why g'(c) in Definition 5.2.1(Differentiability) could have been given by g'(c) = limh→0g(c + h) − g(c)/hIf A is an adjoint operator on H=K2 and for x=(x(1), x(2)), we define A(x)=(x(1)-x(2), x(1)+x(2)), how do we derive that A*(x) = (x(1)+x(2), -x(1)+x(2))?
- Suppose that f1 : [0, 1] → R and f2 : [0, 1] → R are continuous everywhere and that f1(0) < f2(0) and f1(1) > f2(1). Show that there exists a point c ∈ (0,1) such that f1(c) − f2(c) = 0.16. The set S = { x∈R: x2 - 4<0} with the usual metric is .......................... A. Compact. B. Connected. C. Not connected. D. Sequentially compact.Let (+) and (Y ,A) be two topological spaces. Let # be a base for A. Prove that a function f :(4’.r)——({Y ,A) is continuous if and only if the inverse image under ¢ of every member of / is a r- open Set
