Assume that Kis a non-empty closed set In Banach space V and that T:K → K....
3. (1 point) Let (X.11 . ID be a Banach space. K C X be a closed subset and Assume that D40. Prove that the above equality holds true if and only if
3. (1 point) Let (X.11 . ID be a Banach space. K C X be a closed subset and Assume that D40. Prove that the above equality holds true if and only if
Topology
C O, 1 and be the supremum norm (a) Prove that (X || |) is a Banach space. You can assume that (X, | |) is a normed vector space (over R) |f|0supE0.1 \5(x)|.| 4. Let X C (b) Show that || |o0 that the parallelogram identity fails.] on X is not induced by any inner product. Hint: Check for all E[0, 1]. Show that {gn}n>1 (0, 1] BI= {gE X |9||<1} is a compact (c) For every 2...
Assume that there are two non-empty closed subsets S1 and S2 of the state space of a Markov chain. Show that this Markov chain is not irreducible ( that is, the state space is not irreducible)
(3) (10 points) Let H and K be non-empty subsets of a vector space V. The sum of H and K, written H + K, is the set of all vectors in V that can be written as the sum of two vectors, one in H and the other in K: that is H + K = {W EVw = u + v, for some u E H and v EK}. Show that if H and K are subspaces of...
1. (a) Let d be a metric on a non-empty set X. Prove that each of the following are metrics on X: a a + i. d(1)(, y) = kd(x, y), where k >0; [3] ii. dr,y) d(2) (1, y) = [10] 1+ d(,y) The proof of the triangle inequality for d(2) boils down to showing b + > 1fc 1+a 1+b 1+c for all a, b, c > 0 with a +b > c. Proceed as follows to prove...
4·Let A and B be non-empty subsets of a space X. Prove that A U B is disconnected if A n B)U(A nB) 0. Prove that X is connected if and only if for every pair of non-empty subsets A and B of X such that X A U B we have (A B)U (An B)O.
lemma 13
Corollry 12(Sequential ceriterion of a closed set) Let (M. Corollary 12 (Sequential criterion of a closed set) Let (M,d) be a met- ric space. A set S C M is closed if and only if for every sequence (xn) in S that converges in M, the limit of the sequence also belongs to S. Lemma 13 Let (an) be a sequence in (M, d). Let a M. Then, a is a limit point of (ra) if and only...
Problem 9. Let V be a vector space over a field F (a) The empty set is a subset of V. Is a subspace of V? Is linearly dependent or independent? Prove your claims. (b) Prove that the set Z O is a subspace of V. Find a basis for Z and the dimension of Z (c) Prove that there is a unique linear map T: Z → Z. Find the matrix representing this linear map and the determinant of...
Al. Let E be a non-empty set and let d:ExE0, oo). (a) Give the three conditions that d must satisfy to be a metric on E. (b) Ifa E E, r > 0 and 8 0, give the definition of the open ball BE(a) and the closed ball B (a) n-p) closure point of A. Hence, say what it means for A to be a closed subset of E 2 c) Say what it means for a sequence () in...
7. Let A, , An be non-empty subsets of a finite set Ω. If 1 k n and Ek is the set of elements in Ω which belong to at least k of the Ai's show that Pal i-1
7. Let A, , An be non-empty subsets of a finite set Ω. If 1 k n and Ek is the set of elements in Ω which belong to at least k of the Ai's show that Pal i-1