2.) Let (S, d) be a metric space and suppose that ρ : S × S → R is defined by ρ(x, y) = d(x, y) 1 + d(x, y) for all points x, y ∈ S. Prove that (S, ρ) is a metric space, that it is bounded and that ρ(x, y) ≤ d(x, y) for all x, y ∈ S.

Q: 1. Suppose that (X, dx) and (Y, dy) are metric spaces and f: X →Y is a function. For a, b e X,…

A:

Q: 1. a) Let (x, d) be a metric space. Define a flow on (x, d). b) Let (x, {phi_t}) be  a flow on a…

A: The objective of the question is to define a flow on a metric space, identify a fixed point of the…

Q: Let X be an infinite set and let d be the discrete metric on X: {: 0if p=q 1 if p + q. d(p, q) = (a)…

A:

Q: 3) A vector is given as A = x(z+2)â,-()x+1)â, +(y+2)â. Prove the Stoke's theorem for the geometry…

A: From the below figure, ∮cA.dt=∫ABA.ds+∫BCA.ds+∫CDA.ds+∫DAA.ds Here ax=i, ay=j, ak=k Since…

Q: Q2:(a) In Euclidean metric space (R, 1. 1),if A = {y € R: y = 2 cos (2x); x € R} then fined the…

A:

Q: Exercise 4. Suppose (Y,d) is a complete metric space. Consider XC Y as a metric space (X,d) with the…

A:

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: 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: Consider 2-dimensional hyperbolic space with metric ds ds'+dy and y > 0. The length of a line…

A:

Q: 2. Let A = d(x, y) do-metric (i.e. d»((x, a), (x', a')) = max{|x – x'|, d(a, a')}). Give examples of…

A: Discrete metric: Let X be a nonempty set. The discrete metric is defined as follows, dx,y=1 if x=y0…

Q: Consider a set A and a function d: A × A → R that satisfies: • d(x, y) = 0 ⇔ x = y; • d(x, y) = d(y,…

A:

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

A:

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.…

A:

Q: Let (X, || ||x) and (Y, ||- ||y) be the normed spaces. Let XOY := {(x, y): xe X;y ≤ Y} denote the…

A: We know that a norm  on a vector space  is a real valued function on , which has following…

Q: Let A be a non-empty subset of the metric space (X,d). show that |d(x, A)- d(y, A)| < d(x, y)For all…

A:

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: 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 that H=x∈ℝ2:x2 + xy + 3y2 = 3 here we have to show that H is closed subset of ℝ2

Q: Let (X, d) be a metric space. Define f: Xx X → R by f(x, y) = = that f is a metric on X. d(x, y) 1+…

A:

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…

A:

Q: 1. a) Let (x, d) be a metric space. Define a flow on (x, d). b) Let (x, {ϕt}) be a flow on a metric…

A: The objective of the question is to define a flow on a metric space, identify when a point is 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: 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: Let (X, d) be metric and PXIR space, dosy) I+ dexy) a de fined by qusN pusy) = ニ show that p is…

A:

Q: =rcise 7. a) For any metric space show that B₁(x, r) C Int (Ba(x,r)). ind an appropriate metric…

A:

Q: 3.4 Give the definition of a separable met rie space (S, d). 3.5 Show that the metrie space (R, d),…

A:

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

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: d(x,y) 1 - Y -

A:

Q: i) Pe(x) = 0 iff r € Ë. ii) Prove that pe is uniformly continuous.

A: i Let E is the non-empty subset of a metric space X,d the distance from x to E is defined as…

Q: 5. Let X = R? and define d, : R? ×R² →R by d (x,,y,),(x,,y2)) = max {x, – x,|-\y, -y}} . 13 a)…

A:

Q: 6. Let S = R — {0} = (-∞, 0) U (0, ∞) with the usual metric. Show that (-∞, 0) is a closed subset of…

A:

Q: 4. Let {Xn, dn fnɛN be a collection of metrič spaces of D(x, y) = sup{d,(xn, Yn)/n | n E N} a metric…

A: Here we show positivity, symmetric and triangle inequality one by one.

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…

A:

Q: Let V be a 5 dimensional space, and V: V → V linear. Can we have NullT = RangeT? And if the…

A:

Q: Q2: Let (R,7) be the usual topological space f: (R,r) (R,7) fx? +1 :x 21 st. Ar) =-x ;x<1 . Isf…

A:

Q: Q2: (a) In Euclidean metric space (R, 1. 1),if A = {y € R: y = 2 cos (2x); x € R} then fined the…

A:

Q: Consider 2-dimensional hyperbolic space with metric ds = dr'+dy and y > 0. The length of a line…

A:

Q: In space coordinates, the xy plane consists of all the points of the form O (0, y, z) O (X, y, z) O…

A: