OO Let l x.. {x,} ,x, e Vi=1,..00; x be the Banach space х:х— i=1 of absolutely summable sequences of real numbers. Consider the nonnegative orthant in this -el,:x={4} Ax20 Vi= 1,) 1 space, i.e. the set Vi=1,0. Is a cone? Is a set with an interior?

Q: (iv) If a sequence {n} in X converges, it converges to a unique point. (iv) Every convergent…

A:

Q: Find an example to each of the following: a) Sequences {Xn} and {Yn} such that they are both…

A:

Q: Suppose that Collection of. UA iGP is A. (CEP) set Open oppe Open. is Then S

A: We have to prove arbitrary union of open set is open.

Q: (ii) for every singleton set A, m* (A) = 0.

A:

Q: 2/7,545

A:

Q: let M be the lebseque measure on B IR) Calcutate (1) Mes), where S = Where S= { Yn t h+3 i) M( n…

A: We know that measure of countable set is '0'

Q: 1.85 True or False: Finitely many of the open sets in the collection { #) |- wa} * 3r 2' 2 would…

A:

Q: Which 4-tuples a 0<a<b<c<d<6}. are int the relation {(a, b, c, d)|a, b, c, d are integers with

A: We need to identify the tuples of the integer relation .Let .

Q: Let A and B be finite sets with |A| = m and |B| = n. a) How many functions are there from A to B? b)…

A:

Q: Establish formally, whether the following set is open, closed, or neither closed nor open (justify…

A: We will use the basic knowledge of set theory and point-set topology to answer this question…

Q: If it is possible to label each element of an infinite set S with a finite string of keyboard…

A: As per the guideline, I am supposed to solve only first question from the given problems.Given that…

Q: 1 Let B be the set of all bounded sequences of real numbers and define the function d: B x B → R by…

A: we have to solve given problem:

Q: 7. Let Fa be a closed set for k=1,2,..., n. Show that Fk is closed. (Show that a fin and n 1 k=1

A:

Q: of the cantor Show That measure set C is Zero.

A:

Q: Tom wants to invest $500. He invests in a company that predicts his investment will double in value…

A: Initial investment is $500 and the money gets doubled in 3 years.

Q: Let K be a bounded and closed set in a metric space M, K 0. Prove that inf K and sup K are in K.

A: Let K be a closed and bounded set of a metric space M. To Prove: SupK∈K, and InfK∈K.

Q: Let B be the set of all bounded sequences of real numbers and define the function d: B xB +R by d(x,…

A: In the above question it is given that B is the set of all bounded sequences of real numbers and a…

Q: dafine Let A and B n@t be the cardinality f a set 3. It n CA-B) = 18, n CAUB) = 70 nCAnB)= 25, be…

A:

Q: exercise 1.4. Consider Z/24Z. Determine the set {5 . 1, 5 - 2, ...,5 · 23} =? where z is the class…

A:

Q: 2. Let ECR. Show that E is Lebesgue measurable if and only if there exists a sequence of open sets…

A:

Q: Given a finite set S, find all possible values of |S] so that the set {(a, b) | (a E S) A (b e S) ^…

A:

Q: Let X And Y be two distrecte spaces. Then X is homeomorphic to Y if and only if X and Y are both…

A:

Q: let f and g be two bounded read - valu ed functions on the set ASR Showthat sup CPCX) +g X))s…

A:

Q: Which of the following sets is closed in (R, Tu)? O a. { ne N}U{2} 72 2n O b. Q-N c. O c. [2,5) U…

A: Determine the meaning of ℝ, τu. Now, go with each option and check which is correct.

Q: 2) Explain how you count the number of elements in the set A = {n EN| n <500 and gcd (n, M) = 1},…

A:

Q: Prove that the following statement is true for all real numbers x, y ≠ -1, by using a sequence of…

A: y= 1-x/1+x iff x=1-y/1+y proved

Q: Let A = {n€ Z| |n|<3} and the set B = {m/k[m€{2,6}, kE {1,2}}. Write the elements of set ANB in…

A:

Q: Let S, T be any subsets of a universal set U. Prove that (ST)º = S¢ µTº.

A:

Q: Compute the supremum and infimum of the set A = {n/(2n+1) : n ∈ ℕ}

A: We have to find the supremum and infimum of the following set A =n(2n+1):n∈ℕ

Q: In the following list, select all the correct statements, if (4,) is a poset and for any elements…

A:

Q: (Q × {-T, 1})U{-3n, –27}UZ is O an empty set O an uncountable set O a countable set

A:

Q: A = {n EU | n is odd} and B = {n € U | 2n + 1 is a prime number}. Compute the size of the following…

A: Given Information: U=1,2,3,4,5,6,7,8,9 A=n∈U | n is odd B=n∈U | 2n+1 is a prime number

Q: 9. Given an infinite collection An, n = 1, 2, ... of intervals of the real line, their intersection…

A: