Example (3): Let (M, d) be a metric space. Define a function e: Mx M→R by: e(x,y) = Min{1,d(x y)}; for any x,y E M. Therefore (M, e) is a metric space.

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: (X,e) Let be a metric space The diky-k.glry (KEIR) function defined from XxX to IR is a metric on X,…

A: ρ(x,y) be a metric on X ,then it satisfy all the properties of metric  now d(x, y)=kρ(x,y)   k∈R…

Q: Q7:- Find the limit of the following vector functions:- (1) Lim (t²i+ 2tj) (2) Lim(+1i, (3) Lim (ti…

A: We have to find the limit of the vector function.

Q: Let V be an n-dimensional vector space. Define f:V \rightarrow \R by f(v) = /v21. Let p,q \in V.…

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Q: D: Remind that the set F of all functions from R to R is a vector space. 3. Prove that F = 0 E

A: Here the set F of all functions from ℝ to ℝ is a vector space. It has to be proved that F=O⊕E.

Q: 2. (a) Let ds² = v du² − du dv + 2 dv². (a) Find the largest subset U of R² for which this…

A: (a) Given metric:ds2=v du2−du dv+2 dv2Comparing g11=v , g12=g21=−12 , g22=2Given metric will be…

Q: Let f function from (R,d1) to (R,d2) defined as d(x) = x, where d, is the usual metric and d2 is the…

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Q: Let L:R R be a linear operator such that L X2 = X1 then Ker(L) spanned by: -> Select one: a. (e2,e3)…

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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: (a) Define a metric space (X, p). (b) Let p: R+ → R+ be a function defined by ln (2). Prove that p…

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Q: 2. (i) On R², consider the metric d(x, y) = 2x1-9₁ +4₂-9₂- In this metric, sketch the closed ball B₁…

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Q: Sketch the space curve. Vector-Valued Function r(t) = -ti + 5tj + 2tk O Z -2 XX k k Z 5 5 5 2 2…

A: Given Data: rt=-ti+5tj+2tk   0,1 To Find the length of the given interval.

Q: Verify that the discrete metric is actually a metric.

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Q: (a) Give an example of a metric on R2. (b) Is the set of all points (x, y) in R² such that y = x²…

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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.…

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Q: AEsis a subset of a vector space V (F) then show that L (L (S)) = L (S).

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Q: Let V ((x,y): x 20) with standard operation on R2. Is V a vector space?

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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+…

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Q: Sketch the space curve. Vector-Valued Function r(t) = -ti + 6tj + 5tk 2 -2 5 0 y Z Z 2 5 Find its…

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Q: Let (X,d) be a metric space. Prove that the function d' :X x X →R given by d (x, y)= min {1,d(x, y)}…

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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)…

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Q: 2. Let (X, d) be a metric space and y E X. (i) Prove that the closed ball B[y,2] is a closed set in…

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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).

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Q: Let (x, g) be a metric space. Is the function d (x, y) = k.g (x, y) (k € R) defined from XxX to R a…

A: Let X,g be a metric space . Given that function X×X to R defined as  dx,y=k·gx,y where k∈R A metric…

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: Sketch the space curve. Vector-Valued Function r(t) = -ti + 5tj + 2tk 5 ** 11 0 Z 2 2 Interval [0,…

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Q: Ex3 i) ii) Let T: C ([0,1]) → R f → f(0) If C ([0,1]) is equipped with the sup-norm, show that T is…

A: As per the policy, we are solving first question. Please repost it and specify which question is to…

Q: A. Show that T is a topology in X and Show that B,(x) is open set with respect to t for all r > 0

A: This is a problem of Topology.

Q: Let (X, T) and (Y, T1) be two topological spaces and let f be a continuous mapping of X into Y. If…

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Q: Exercise 6. If ||·|| is a norm on the vector space E, show that the formula d(x, y) = ||y — x||…

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Q: the space curve (t) = (2t, t², 2t)

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Q: Match the space curves in the figure below with the following vector-valued functions: O A. r(t) =…

A: See that options ,  A )  r ( t ) = < 1 , t , t >  If we consider the first two coordinates ,…

Q: Let f:R2→R be the function defined by f(x,y)=−(x−4)2−(y−4)2. Consider the region A of R2 given by…

A: here we use the concept of compactness and continuity in ℝ2. Her we need not to find the maximum and…

Q: the metric space symmetry condition of the discrete metric, I find myself writing that if d(x,y)=0…

A: Let (X,d) be a metric space, where d is the discrete metric. Then, for any x,y∈X, we have: d(x,y) =…