Problem & (3 points). Complete the definition of a metric space by filling in the three blanks: A metric space is a pair (S, d) where S is a set and d: → [0,00) is a function such that

Q: If X= R and d be the discrete metric Then the set of natural number (N, d) is complete no decision…

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Q: 1. (4 points) Show that the space (C([a, b]), ||· ||∞) is a Banach space.

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Q: (b) Is (R2,d) a metric space, where d((x.y1).(x2.y2)) 1x1-x2l; for all (x, , y1 ), (x2 2) E R2 ?…

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Q: Suppose A, B are non-empty, disjoint, closed subsets of a metric space (X, d).

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Q: A. Let H be the set of all points (x, y) in ℝ2 such that x2 + xy + 3y2 = 3. Show that H is a closed…

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Q: A completely regular space is also T1-space 1 True 2 False

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Q: QUESTION 5 The set R-{0} with the usual metric is connected. O True O False

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Q: DATE MONTH 9- verify that l ell max alt)| defines. t elasb] %3D the Space cla,b]. morm on

A: Norm: A norm· is mapping defined on vector space V which satisfies following properties: 1. x ≥ 0,…

Q: the quotient space of a T0−space necessarily T0

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Q: Let M = 0,2) and d be the usunl metric, then dis not complete O True O False

A: False

Q: Q7. Assume that (R,T) is the usual Euclidean topological space. N = {1,2, 3, ....} C R. Determine…

A: There is no interior point of N.

Q: . Let A and B be two connected subset

A: “Since you have asked multiple questions, we will solve the first question for you. If you want any…

Q: (a) For the following subsets in an indicated metrie space detennine the interior, boundary points…

A: See the attachment.

Q: Let X be the space of bounded and closed linear segments on the positive part of the real line, so…

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Q: Q8. Assume that (R²,T) is the usual Euclidean topological space. D = {(x, y) E R², y = x} C R².…

A: Given below the answer in detailed

Q: Using the fact that A/B = An Be, show that in a metric space, bdy(A) = bdy (Aº)

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Q: A. Consider R3 with the metric d((X1, X2, X3), (V1, Y2. Y3) = |X1 - Yıl + |x2 - y2l + [X3 - Y3l. i.…

A: This problem is related with metric on ℝ3. Given that                   dx1,x2,x3,y1,y2,y3 = x1-y1+…

Q: If X= R and d be the usual metric Then the set of natural number (N, d) is not complete no decision…

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Q: Let X be any nonempty set, and define (o if x = y. 11 if ry. Prove that (X. d) is a metric space.…

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Q: Example B Let M=IR², and let x = (x₁, x₂), y = (y₁, y₂). Then d₂ = √√(x₁ - y₁)² + (x₂-Y₂1²1 is the…

A: Let X be a vector space. A real valued function d:X×X→R is a metric if it satisfies For all…

Q: A. Let H be the set of all points (x, y) in R² such that x² + xy + 3y² = 3. Show that H is a closed…

A: To Prove: H=x,y|x2+xy+3y2=3 is a closed set in ℝ2. To Examine: H is bounded or not.

Q: M is an infinite set. If x,y ∈ M, explain the relation f below            {  1 , if x = y   f(x,y)…

A: We have to solve given problem:

Q: The set [0,4] with the usual metric is sequentially compact. O True O False

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Q: Which of the following sets is not closed on the real line with the usual metric? (i) The set of…

A: Option (iii) the set of rational numbers is not closed on the real line with the usual metric.

Q: 16 + 3 the following matrices festSIAM #71 47 purन

A: Topic = Matrix

Q: = 1. Let (X, d) be any metric space. Difine p: X X X → R as p(x, y) d(ry). Then (X, p) is a metric…

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Q: r', where i =

A: As per rule, allowed to answer first question and post the remaining in the next submisison.

Q: (a) Provide an example of a contraction between metric spaces which has no fixed point.Justify all…

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Q: If X= R and d be the usual metric Then the set of irrational numbers (Q', d) is complete no decision…

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Q: Determine the dimension of the space spanned by the set

A: Given,               S=12-13, 1002, 28-48, 1111, 3306

Q: Let1. (X, M,) be a space with measure. Show that for arbitrary A, B E Mthere is an equality: /x.du =…

A: We are given that (X,M,μ) be a measurable space and A,B∈M. We need to prove that  ∫AχBdμ=μ(A∩B).

Q: Remark. Compactness of a set ECX depends on the metric p. Example 3.16. In R with the Euclidean…

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Q: 6. Which of the following spaces are isomorphic to each other? R, P3, M2x2, R*

A: We will use basic of Isomorphism.

Q: pls explain to me step by step and pls write clearly. thanks

A: To verify that the contraction mapping condition is NOT satisfied using the given functions f(t) and…

Q: Prove that Let X be a metric space such that X E1 U E2. If E1, E2 are complete, then X is complete.

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Q: 6. Let A and B be two connected subsets of X and An B + 4. Show that AUB is connected. In…

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Q: Find the solution of the initial value problem y" + 4y' + 5y = 0, Y (²) = 0, y' (²) 7. y(t) = How…

A: The given initial value differential equation is y''+4y'+5y=0, yπ2=0,y'π2=7 The objective is to…

Q: Show that the Euclidean metric is a metric( i.e. verify that the function d satisfies the three…

A: Here we have to show thatt Euclidean metric is a metric.

Q: Let (X, d) be a metric space, and Y be a non-empty subset of X. (i) Equip Y with the distance…

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Q: Evaluate the triple integral. 8x dV, where E is bounded by the paraboloid x = 2y2 + 2z2 and the…

A: The objective is to evaluate the triple integral ∫∫∫E8xdV, where E is bounded by the paraboloid…