Example (4): Let (M, d) be a metric space. Define a function e : M X M → R as: e(x,y) d(x,y) 1+d(x,y)' vx,y e M. Then, (M, e) is a metric space.

Q: Let V be the vector space of functions from R into R. Show that f,g,h EV are independent where: f(x)…

A:

Q: 4. Check that the following are vector spaces: (a) RN. 55 / 600 (b) CN. (c) C(N), where 2 is a…

A: A non-empty set V, +, · is said to be a vector space over a field F together with operations of…

Q: Which of the space curves below describes the vector-valued function r(t) = (t + 15, e0.08t cos t,…

A: The given parametric equation of the curve is  where  is the parameter.To determine which of the…

Q: Show that X'= B(X, K) is a Banach space.

A:

Q: Is the set V = {(x, y) x >0} where (x1, y₁) + (x2, Y₂) = (x₁x2, y₁ + y₂) and c (x, y) = (x,cy) a…

A:

Q: Q4) Let C(R) be a normed space such that || f -sup{ Ifl,x E R}and T: C (R) -R define Tf)-If I. Is T…

A:

Q: Let V be an n-dimensional vector space. Define f: V→ R by f(v) = |v|". Let p,q E V.Find d fp(q).

A:

Q: Find the domain of vector r(t) and find the limit as t goes to 0 of vector r(t)

A: According to question,Given vector,rt = 2-t, et-1t, lnt+1

Q: Let V be an n-dimensional vector space. Define f : V \rightarrow \R by f(v) = |v2|. Let p,q \in V.…

A:

Q: let (x,d) be a metric space. define a flow on (x,d)

A: Step 1:The concept of flow is often used in the study of dynamical systems, where X represents the…

Q: Q1. (a). Let V = R and define addition and scalar multiplication by: (a, b) e (c, d) = (a +2c,…

A: See the detailed solution below.

Q: Is the set V = {(x, y)|x > 0} where (x1, Y1) O (x2, Y2) = (x1x2, Y1 + Y2) and c O (x, y) = (x°, cy)…

A:

Q: Let V be an n-dimensional vector space. Define f : V→R by f(v) =|v|². Let p, qe V. Find d fp(a).

A:

Q: iii) (f x g) xh is differentiable at t and d dt 7 {(f x g) x h} xgx h + x dt dg xh +(fx g) x dt dt

A:

Q: Q4) Consider the space (X.Tu) and let A = [0,2) U (3) U (4,7). Find d(A), A, Aº, ext(A) and b(A).

A:

Q: * Let (X, d) and (Y, e) be metric spaces. Show that the function f: X xY X defined by f((r, y)) = x…

A:

Q: Let E and F be two vector spaces with the same finite dimension dim(E) = dim(F) For every linear map…

A: Follow the steps.

Q: Consider inner-product = | f(x) g(x) dx defined for vector space C[-1, 1] , then for the function…

A: The solution is given as

Q: (b) Let X bę a normed linear space. Let Вх, г) %3D (у є Х: || x-у || <r). Show that B(x, r) = {y e…

A: To show - The ball B¯(x, r) is closed. Indeed, for any given y ∈ M \ B¯(x, r) we set R = d(x, y)…

Q: Sketch each of the following sets and decide if they are open sets. Give brief reasor for your…

A:

Q: B. Is C a real vector space? Explain.

A:

Q: Which of the following pairs of vector spaces are isomorphic? Justify your answers. (a) F³ and P3…

A: Two finite dimensional vector spaces are isomorphic iff they are of the same dimension.

Q: If V is a vector space then show that cV=V.

A: