Exercises: 1) In a metric space (M,d). show that both Ø and M are open sets in M.

Q: Is the euclidean space R³ is a complete metric space

A: We have to check whether euclidean space R³ is a complete metric space or not .

Q: Let S be the set of all sequences of real numbers (xn) for which x < 0, n=1 that is, the series x…

A: Given distance function is

Q: Find the solution set for the equation AX = b where [o -1 0 21 O 27 a) A = -1 10 1 Deduce from this,…

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Q: Let R be the set of real numbers and d: R x R → R be defined as follows. d(x, y) = 3|x8 – y®|. %3D…

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Q: Let R be the set of real numbers and d: R x R → R be defined as follows. d(x, y) = 5|x10 - y1º|.…

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Q: (E, Q4: state and prove L is a normed space ? e a ne. Vila a convex set.

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Q: Given a metric space M with metric d, verify that any e-ball is an open set.

A: Given That : Metric space M with metric d . To Verify : Any ε-ball is open set.

Q: B. Given a metric space M with metric d, verify that any ε-ball is an open set. (Please If possible…

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Q: C. Show that a set A in R² is open in the Euclidean metric it is open in the max metric.

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Q: Q. Let (X, B(R), µ) be a Borel measure space. If An B = 0, BnC = 4 µ(ANC) = 0 Determine u(AUBUC).

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Q: Determine the diameter of the set U = (9, 14) U (22,31) in the metric space R with metric being the…

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Q: B) Let X = R be a metric space and consider the following subsets of X, %3D 1 1 S = (0,1), S2 %3D…

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Q: 1. Show that the real line is a metric space.

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Q: Given a metric space M with metric d, verify that any e-ball is an open set.

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Q: Prove that (R", II-|) is complete metric space.

A: This is a problem of Functional Analysis.

Q: Show that (Cod) is metric space

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Q: Let V = (a, b, c, d), such that a = 2b − c, b = 2d − 3c, and d = a − b + c. Find the dimension of V…

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Q: Let A = {a | a is a real number} and B = {1, 2, 3}.  Sketch the given set in the Cartisian plane.…

A: real number = rational number + irrational number rational numbers = { 1,2,3/5,7/9....} irrational…

Q: Q1:- Let A, B, C, D, E be subsets of the Euclidean space R². Find their boundary, their interior,…

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Q: Show that a set A in ℝ2 is open in the Euclidean metric ⇔ it is open in the max metric.

A: Answer given below.

Q: B) Let X = R be a metric space and consider the following subsets of X, 11 S, = (0,1), S2 %3D p....…

A: Given  S1 = (0,1) and S2 = 1,12,13,...,1n,...,0

Q: Consider the sets N = R? \ {(0, y) E R² | |y| > 0}, %3D L = {(x, y) E R² | |¤| = |y|}, and let di,…

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Q: 2. Consider a metric in a D-dimensional space. Simplify the following expressions, i.e. the result…

A: note :Since you have posted question with multiple sub parts, we will provide the solution only to…

Q: Given a metric space M with metric d, verify that any E-ball is an open set.

A: Given that M is a metric space with metric d. Let Ba,ε be an ε-ball with center at a. That is:…

Q: the euclidean space R³ is a separable metric space. true or false?

A: A subset M of a metric space X is said to be dense in X if M=X.X is said to be separable if it has a…