Let X = (0,00). -> a) Show that X is not complete with the standard metric d: X2 → [0, 0o) defined by d (x, y) = |x - yl b) Use #1 to find a metric p on X so that X is complete with respect to p.

Q: Exercise 1. Let (X,dx) be a compact metric space. (a) Verify that dy(f, g) = E 2-"dx(f(n), g(n)) nƐN…

A: let X,dX be a compact metric space dYf,g=∑n∈ℕ2-ndXfn,gn defines a metric on Y=f:ℕ→X to show that Y…

Q: Let y e R. Determine if the following are metrics on R or not: d1(x,y) = sqrt(abs(x-y))…

A: A set X and a function d together called a metric space if the function d:X×X→ℝ satisfies the…

Q: Suppose (X, d) is a metric space and let r> 0. Show that p: X2 → [0, c0) defined by p(x, y) = r·…

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Q: Determine whether d is defined on R. Is it metric on R? d(x, y) = 13x - yl Also: if metric, show…

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Q: a) Use Green's theorem to compute f F dr where c is a circle of radius R oriented CCW and F(x, y) =…

A: Green's theorem:∫C[Pdx+Qdy]=∫∫T∂Q∂x-∂P∂ydxdy where T is the region bounded by a simple closed curve…

Q: 1. Use Green's Theorem to evaluate |F. dr, where F(r, y) = (3y- esin, 7x + Vy + 1) and C is the…

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Q: 2. Prove that if d is a metric for X, then another metric d' for X is given by if a ≤ b " if b ≤ a…

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Q: 3. Let d₂(x, y) = |arctan(x) — arctan(y)| for x, y € R. (a) Show that d₂ is a metric on R. (b) Are…

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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: F. dS, where F(x, y, z) = z²xi + 2) + (x2z + y2)k and S is the top half of the sphere x² + y² + z? =…

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Q: Let de(x, y) := |x — y|, d₁(x,y) := for x,y E R. Show that (a) d₁ is a metric on R; (b) dẼ and d₁…

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Q: For X = (x₁,...,xn), Y = (y₁, ..., Yn) € Rn, define n d(X,Y) := Σ |xi - vi| i=1 (a). Show that d is…

A: Given that, are two vectors.The defined function is: To check the d is matrix space on , have to…

Q: Define d (x,y)= (x²y²1, x,y ER. Is d a metric on IR?

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