- - 1. For any x, y € R², define d₁ (x, y) = max{|x1 — Y₁|, |x2 — Y2|}. Prove that d₁ is a metrics on R². 2. For any x, y ER², define d₂ (x, y) = |x1 − Y1| + |x2 − y2|. Prove that d2 is a metrics on R².

Q: Ques. 1. Let N be a linear space and d be a metric on N such that d(x + z,y+ z) = d(x,y), x,y € N…

A: I have proved all three conditions gor norm

Q: Let X = [0, 1] and d be the usual metric, d(x, y) = |x- y| %3D O True O False

A: TRUE

Q: 3- In a complete metric space (X, d), If A C X such that, A = A then A is complete as a sub space of…

A:

Q: (c) If X= R, with r={ R, ¢, Q, Irr}. Show that (R, t) is a compact space.

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Q: = 0,1] and d be the usual metric, d(x, y) = a -y %3D en d can not induces a norm on X True False

A: the usual metric d(x,y)=|x-y| usually induces the norm.

Q: Let X=R2 and let d be the Euclidean metric.  Define d3:  R2 x R2 to R by d3(x, y) = min{1, d(x, y)}…

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Q: Q3: (a) In the metric space (S, d), Vx,y ES. If A = where S = [0, 1]; d(x, y) = |x-yl; A = [0,2),…

A: A metric space is a set together with a metric on the set. The metric is a function that defines a…

Q: Let X = N and d be the usual metric, d (x, y) = |x – yl then d can induces a norm on X True O False…

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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: For every x, y E R, let p(x, y) = |x² – y²\. Is p a metric on R?

A: Hello. Since your question has multiple parts, we will solve first question for you. If you want…

Q: Let x = (13, −20, 15), v1 = (1, −2, 3), and v2 = (−1, 1, 1). Then the set {v1, v2} is orthogonal. It…

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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: (b) Let (M, d) be a metric space. Define a function e: MXM-R by e(x, y) = min {1,d(x, y)}; vx,y E M.…

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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: Let (X, d) be a metric space, x E X, e > 0, and E = {y Ex: d(x, y) ≤ e}. Prove that E is closed.

A: In this question, we need to show that set E is closed. Equivalently it is sufficient to show that…

Q: 2. Let d1, d2 be two metrics on X + 6. Show that: d(r, y) = d1(x, y) + d2(x, y).

A: I have proved all four conditions of metric for d

Q: 3: Let a function d: R* x R* → R defined by d(x, y) = E-\xi - yıl Vx = (x1,...,x4), y = (y1,...,y4)…

A: Here we will prove that d is a metric function.

Q: 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: Given : As per the question we are given the following bivariate function : x2 + xy + 3y2 = 3 And a…

Q: QI/ a) Suppose that f: [0, co) y is a metric space. Show that d(x, y) = fCo(x, y)) is a metric on X.…

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

A:

Q: Let X =  R x R  x R = R3 and define the metric d: R 3 x R 3 to R  by d((x1, y1, z1), (x2, y2, z2)) =…

A: Consider X=R×R×R=R3 Define the metric d:R3×R3 to R  by d((x1,y1,z1),(x2,y2,z2))=x1-x2+y1-y2+z1-z2…

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: If M and N are metric spaces, the Cartesian product M x N is again a netric space with the distance…

A: We have given that , M and N are metric spaces. We have to show that , M × N is again a metric space…

Q: Let F = Vf and f = 5x²y - 5z. Calculate F. dr for the path r₁ = (t, t,0), 0 ≤ t ≤ 1. (Give your…

A: We will use fundamental theorem of line integrals

Q: Let X=R2 and defined d2:  R2 x R2 to R by d2((x1, y1)) = max{|x1-x2|, |y1-y2|} Verify that d2 is a…

A: Given a function d2: R2×R2 → R, defined by d2x1,y1,x2,y2=maxx1-x2,y1-y2. To verify that d2 is a…

Q: 2. Let (X, d) and (X, d') be a metric spaces such that there is KEN satisfying 1 d'(p, q) ≤d(p, q) ≤…

A:

Q: Let = (1,2). - a. Explain graphically the set S = {(x, y) : (x − 3, y − 1) · ʊ = 0}. b. Draw the the…

A: v→=1,  2 S=(x, y) : x-3, y-1·v→=0

Q: Exercise. Prove that in any metric space (X, p), a closed ball {r E X : p(a, x) <r} is closed.

A:

Q: Consider the following mappings on RXR: * d1(x, y) = e2z-y, d2 = |2x – 2y|, and d3 = VI2x – y|. %3D…

A:

Q: Let d be a trivial metric on a non-empty set X and d₁: XXX → R such that x = y d₁(x,y) = { 2 if x…

A: Given:d1:X×X→R such that  d1x, y=0if x=y2if x≠y∀x , y∈X To Show: d and d1 are equivalent.

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

A: Interior Points: Let (R,d) be a metric space and S⊆R. A point p∈S is called the interior point of S…

Q: d1(x,y) = e*-y, d2(x,y)= |2x – yl and d3(x,y) = /]x – yl %3D %3D O Both d1 and d2 are metrics on R.…

A: We have,d1x,y=ex-yd2x,y=2x-yd3(x,y)=x-yWe will check whether they satisfy the properties of metrics…

Q: Exercise.3 Show that the following the functions d and p on X = R? are metrics or not, such that 2.…

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Q: 1. Let X R². For any (x1, y1), (2, y2) E R², define d₂((x1, y₁), (x2, y2)) := Show that d₂ is a…

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Q: Give the definition of a separable metric space (S, d). Show that the metric space (R, d), where…

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Q: Let (X, d) be a metric space and EX. Prove that if E is compact, then E is bounded.

A: Let be a metric space and . A family of subsets  is called an open cover of  if each  is open and…

Q: Define on R, the metric d defined by d(x, y) = |1+1리 1+ |y| Then (R, d) is not complete. True False

A: Given

Q: Prove that the d1 (“Manhattan”) metric on R2 is really a metric

A:

Q: Let (X. d) be a metric space and define p: XxX by ptr.g)= dx.y) Prove that (X,p) is a metric…

A: Introduction: To prove that (X, p) is a metric space, we need to show that the following three…

Q: Consider the following mappings on RXR: d1(r, y) = e , d2 = |2.r – 2y|l. and d3 = v2r - yl. None of…

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Q: Consider ℝ3 with the metric d((x1, x2, x3), (y1, y2, y3) = |x1 - y1| + |x2 - y2| + |x3 - y3|. i.…

A: To verify that given distance form d is a metric , we have to check for  Identity Symmetry…

Q: Verify that (S, d) is a metric space. 10.2 Prove that d(x, y) < d'(x, y)s/2 d(x, y) for all points'…

A: To prove dx,y≤d'x,y≤2dx,y for all points x,y∈ℝ×ℝ where d is the usual metric and d' is the…

Q: In a certain geometry, define the distance d(P₁, P₂) between two points P(x, y,) and P(x, y₂) as…

A: We will use the basic knowledge of metric spaces to answer this question completely and correctly.