4. Suppose that (X, dx) and (Y, dy) are metric spaces, and define distance on X × Y by Prove that (Tmyn) → (z, y) in X × Y if and only ifFm → z in X and Un → y in Y. 5. If f:X → Y, the graph of f is the set G = {(x, f(x)) : x E X} X × Y. Suppose that f is a mapping of a compact metric space X into a metric space Y. Prove that f is continuous on X if and only if its graph G is compact. Hint: Use Exercise 4, and for the - part, also Exercise 3 applied in G.

Q: Let L be the line in R2 through the points [:] } Find a linear functional f and a real number d such…

A: First, we will find the equation of a line given two points.

Q: b) Is there a linear map T: V → W, where dim V = 3, dim W = 4, rank T = 2, and dim ker T = 1?

A:

Q: 3. This question concerns the L metric d and the L¹ metric e on a) Calculate d(f.g) and e(f.g),…

A: Consider L∞ metric d and L1metric e on C[0,1]. (i) To Find: d(f,g) , e(f,g) for  f(x)=1+x , g(x)=x2.…

Q: \si E2. is a metric on 1. Prove that d[(s), (t)] = 1=-00

A: To prove, ds,t=∑i=-∞∞si-ti2i is metric on ∑2. A metric space is a set X together with a function d…

Q: 2. Find a formula for curl(curl(u)) in terms of (x1,02, X3) and (e, e2, e3) if u = uje1 + Uze2 +…

A: We will use the basics of vector calculus to answer this question in detail. Below is the…

Q: Let F: X Y and G:

A: Given: Let F : X→Y and G : Y→Z be the two non-constant holomorphic maps between Reimann surfaces.

Q: suppose that S = [a, b] x {0} is a closed line segment %3D in R?. Show that H' (S) = b a.

A: Here given as S=a, b×0 is a closed line segment in ℝ2. We have to prove that H1S=b-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: It is estimated that 60% of emails are spam emails. Some software has been applied to filter these…

A: Thanks for the question :)And your upvote will be really appreciable ;)

Q: Let (X, d) be a metric space, and p: X→ IR doxy) I+desy) de fined by puM pusy) = %3D show that p is…

A: Consider a set X and a function defined from it to the real numbers as d : X×X→ℝ. Then, X,d is 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: Solve using method of undetermined coefficients: y" + y' - 2y = x². %3D

A: Solution :-

Q: Suppose that we define a function d that maps R2 to R by d(x,y) = { 0,. if x = y 1, if x is not…

A: Given. Let d be a function with the mapping d: R2 → R defined by, dx,y=0,  if x=y1,  if x≠y   (a) To…

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: 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: Decide which of the following are metrics on X = R2. For each, we let x = (x1, x2) and y = (y1, y2)…

A:

Q: Let V(F) be an inner product space and a, Be V. Show that a= B iff ( α, γ)- (β,)V γε V.

A:

Q: 7. Consider a line in 2-dim space y = ax + b and a given point (ro. 30) not on the line. Please…

A:

Q: 4. Let (X, d) be a metric space and d' be a metric on X that is equivalent to d. Let and be the…

A: Let us prove above result

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

A:

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: 4. Show that the closed interval [0, 1] and the unit circle S¹ = {(x, y) | x² + y² 1} cannot be…

A: Suppose there exist a homeomorphism f:[0, ∞)----->Rthen restriction of f f|[0,∞)\{0} : [0,…

Q: ! 0. JUUTTUU M 10. Compute the distance between the functions f(x) = 5 + x and g(x) = 6 + 3x in…

A:

Q: Define the following a) span {v₁, v₂,v}, b) {v₁, v₂,vis linearly n dent, mation. m c) span {v₁, v₂,…

A:

Q: Find a continuous function. f:D → R (where R is the real number line) such that f((x1, y1)) = f((x2,…

A:

Q: A rental truck costs $45.45 for a day plus 94¢ per mile. (a) If x is the number of miles driven,…

A: Given that, a rental truck costs $45.45 for a day plus 94c per mile.

Q: . Let X be a nonempty set with metric d. Show that the function e defined by e(x, y) = min[1, d(x,…

A: We know that d(x,y) is a metric on X. Hence d satisfies all the properties of a metric.  Now e(x,y)…

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

A:

Q: If T is surjective and V = span(3), prove that W = span(T(3)). Hint: given that span(T(3)) ≤ W, it…

A:

Q: - - 1. For any x, y € R², define d₁ (x, y) = max{|x1 — Y₁|, |x2 — Y2|}. Prove that d₁ is a metrics…

A:

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

A:

Q: 3. Consider the real Cartesian plane with the standard distance function, d = the square root of…

A: Only first individual question is to be answered. Kindly repost the remaining questions.

Q: 1. Let (S, d) be a discrete metric space and p E S. Identify Sı (p), S1 (p), S2(p). 2. Let R have…

A: We will use the definition of metric function to sketch.

Q: Let H = H4 U H, be the class of "positive" and "negative" circles in R² (the "negative" circles are…

A:

Q: 3 Let (Y, p) be a metric space. Give an example metric d on Y such that every finction f:(Y, d)→(Y,…

A:

Q: 12. Let v =|0 and y For what value(s) of h is y in the plane -3 -2 generated by v and v2?

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

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…