metric space is a set M together with a distance function ρ(x,y) that represents the distance" between elements a and y of the set M. The distance function must satisfy ) plz,y)20 and plr,y) 0 if and only if y: (ii) p(x,y) o(y,z); (iii) p(z, y) a(x, z) +a(z, y) for all x, y, z in M. Let (M.p) be a metric space. A mapping T from M into M is called a contraction if for some constant a with 0 sa<1, and all a and y in M I. Prove that the real line R becomes a metric space if we define a distance ρ(x, y) between real numbers a and y by Here, your objective is to show that p(z. g) satisfies the three conditions (i), (), and (ii) above. You may assume without proof the "triangle" inequality for the absolute value function for all a, b in R

Q: 2. An is a set of points (x, y) in a plane such that the sum of the distances between (x, y) and two…

A: We know that, The sum of distances to any point on the ellipse x,y from the two foci c,0 and -c,0 is…

Q: (b) Consider the set R. Prove: (i) VxER, x 0 = 0; (ii) If x y = 0 then either x = 0 or y = 0, or…

A:

Q: 3. Let r>0 and define the closed ball by B,(z) = {y € X : d(x,y) Sr} Show that B,(r) is closed set…

A: Given X be a metric space with metric d. Now define a topology τ on X as follows: τ=B⊆X:…

Q: A(n)................................... is the collection of all points in the plane such that the…

A: We know, A parabola is a collection of all the points in a plane such that the distance from each…

Q: Let z=x+iy ∈ C where x∈R\{1}, y∈R, and |z|=1. If w = (z+1)/(z-1), then show that Im(w) = y/(x-1).

A: Use the following formulae, to obtain the required result. Real part of a+ib=aImaginary part of…

Q: If there exists open ball B (a, r) such that B (a, r) NG= Ø then a is not limit point of G True O…

A: TRUE

Q: Suppose that l is a line. Let H be one of the two half-planes bounded by l. Give a proof that l U H…

A: Answer is mentioned below

Q: f(x,y) = (In(y + x²)) x² + y² D = {(x,y) ERIy2x²-1 and y ≤4-e-*) (b) Observe the set Is D…

A: (b) The given set is D=x,y∈ℝ : y≥x2-1 and y≤4-e-x To find: The given set is bounded or closed or…

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: Two functions u and v of x and y are said to satisfy the Cauchy- Riemann eguations if u, = v, and u,…

A: Thje given functions are,u(x,y)=y(x2+y2) , v(x,y)=x(x2+y2)

Q: If there exists open ball B (a, r) such that B (a,r) nG = Ø then a is not limit point of G True…

A: TRUE

Q: 1. Let F(x, y,z) = (x + y², y + z²,z + x²). Let C be the triangle with vertices (1,0,0), (0,1,0),…

A: Consider the provided question,

Q: Let X and Y be two T, spaces, then X×Y. :is

A:

Q: (v) Prove that if the space (X,T) has the fixed-point property and (Y, T1) is a space homeomorphic…

A: Given: v) To prove the given statement as follows,

Q: 5. Let (X,r) and (Y,A) be two topological spaces. Prove that a function f:(X.r)(Y.A) is closed if…

A: Since you have posted multiple question according to company rule we are supposed to give solution…

Q: of at (x, y, z) = (2, 3, 4). ay Let f(x, y, z) = (x + y)z. Evaluate

A: Given :fx,y,z = x+yz

Q: Q2:(a) In Euclidean metric space (R, 1. 1),if A = {y & R: y = []; x E R) then fined the following…

A:

Q: Q. Assume that (R,T) is the usual Euclidean topological space. Let ([1,6], T1,6) be the relative…

A:

Q: Let {(x, y) E R² : |x| 1}. {(x, y) E R² : 0 < x < y}, {(x, y) E R² : 0 < y <¤ < 1}, U {(7, y) E R² :…

A:

Q: (b) Show that x. y < ||x|| ||y||- [Hint: If x, y # 0, let a = , 6= and use the fact that ||ax + by |…

A: Given that, Euclidean metric d on Rn is defined as d(x,y) = || x - y || for x, y in Rn Part(a): Let…

Q: Q2: In Euclidean metric space (R², 1. 1), is the set A = {(x, y): xy < 1} compact? giving the…

A:

Q: 1. Let F(x, y, z) = (x + y², y + z²,z +x²). Let C be the triangle with vertices (1,0,0), (0,1,0),…

A: by using guass divergence theorem

Q: O Let ( X, T) be topdogical space and A,B CX- ldentify the Condition when a (Int(A)u FrlB) Ul F CA)…

A:

Q: Let, T:R' → R';T(x,y,z) = (x + y+ z,2y+z,2y+3z) 2. Find the dimension of KerT and ImT

A:

Q: Let r, y EN such that y # 0, then y+r#0.

A:

Q: If there exists open ball B (a, r) such that B (a, r) nG=Ø then a is not limit point of G

A: To check whether there exists open ball B(a, r) such that B(a, r) ∩ G= ∅, then whether a is not…

Q: Let r = ⟨x, y, z⟩ and r0 = ⟨1, 1, 1⟩. Describe the set of all points (x, y, z) such that r − r0 =…

A: Given: r=x,y,z and r0=1,1,1 To describe the set of all points x,y,z such that r-r0=2 Solution:…

Q: A(n)............ is the collection of all points in the plane the sum of whose distances from two…

A: It is given that,   A(n) ............ is the collection of all points in the plane the sum of…

Q: ht x and y be normed spaces. Show that a Lincar operator T:Xy is bounded If and only T maps bounded…

A:

Q: (b) Let X = {0, 1} and let (X*,d) be metric space, where d(x, y) = the number of elements of the set…

A:

Q: Let V be a real inner product space, and let u, v, w E V. If (u, v) = 1 and (v, w) = 3, what is (3u…

A:

Q: 3.3 Let r1 and r2 be distinct points in the metric space (S, d). Verify that there are open balls S,…

A:

Q: (йх'ў):(wx2)

A:

Q: A subset S of a plane satisfying incidence and betweenness axioms is called convex if for any two…

A: (a)To prove: - Intersection of convex set is convex.Let us take two convex set (let A and B) and…

Q: pls explain to me step by step. pls write clearly and dont skip any steps. thanks

A: To test the contraction mapping property for the same operator T in two different settings . (C[0,1]…

Q: 28. Bounded by the planes z = x, y = x, x + y = 2, and z

A: Solution:-

Q: ii) True or False: The space R is homeomorphic to the closed interval [-1, 1] with the usual…

A:

Q: Pls explain to me step by step and pls dont skip any step. Thanks

A: To prove that the taxicab distance is actually a metric on the plane

Q: Show that the linear map (x, y)H (x+ y, x- y) from (R?, || - |1,) to (R?, I| - ||is an isometry.

A:

Q: Let, T: R' → R';T(x, y,z) = (x+ y+ z,2y + z,2y +3z) Find the dimension of KerT and ImT

A:

Q: Let X=R and T = {]a, +oo[/a R}U{o,R}, then (X, 7) is a regular space. True False

A: This is a problem of topology.

Q: pls explain to me step by step and pls write clearly. thanks

A: To verify that the contraction mapping condition is NOT satisfied using the given functions f(t) and…

Q: Prove that R" is connected; Prove that [0, 1] × [0, 1] (with the relative topology from R²) is not…

A:

Q: ii) True or False: The space R is homeomorphic to the closed interval [-1,1] with the usual…

A:

Q: Theorem:For all Xiyiz,w EN i) X<y ^ z sw =)x+Z <ytw ii) xsy ^ zw =) x+Z< y tw )x<う tsw 1 Esw うメ+ と x…

A: Solution: We know that for x,y∈ℕ If x&lt;y then ∃k∈ℤ+,k&gt;0 such that x+k=y If x≤y then ∃k∈ℤ+,…

Q: Let R have its usual topology and let Y = {x, y, z}. Define f: R → Y by: %3D x 0 (х, f(x) {y, (z,…

A: Quotient topology: If X is a topological space and A is set and if p:X→A is a surjective map, then…

Q: Let f(x, y, z) = 2x(y+z), x uv, y = v w, and z = w. Find in terms of u, v, and uw.

A:

Q: 1. Let J = {Ø, X,{u},{v,w}} be a topology on X= {u,v,w}. Let J' = {ø, Y, {p}} be %3D | a topology on…

A:

Q: Let V be an inner product space. Then for x, y, z ∈V and c ∈F, the following statements are true.…

A: