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: Ex: Let (X, d) be a metric space and E, andE2 EX Show that whether each one of the following…

A: According to guidelines I solve first 3 subparts so kindly request you to repost the remaining…

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: If X =R and d(x.y)30, then X metric space not metric semi metric

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Q: Example (4): Let (M, d) be a metric space. Define a function e : M X M → R as: e(x,y) d(x,y)…

A: Metric space is the base of topology. A metric space is a function that maps a non-empty set under…

Q: Thm. Let v be an inner product Space then and HyE V жує ||≤|12||· 1191/ further equility is hold iff…

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Q: Thm. Let v be an V be an inner product Space V and then жує | 13 11211. 11111 further equility is…

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Q: For each case, determine whether the metric space is complete or not. Support your answers. (i) Z…

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Q: (c) If X= R, with r={ R, ¢, Q, Irr}. Show that (R, t) is a compact space.

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Q: (3) Let R² and R be endowed with their Euclidean metrics. (a) Show that projections p₁(x, y) = x and…

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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: (1.7) Example: Let P,(R) the vector space of real functions, define by: = f(x)g(x Is inner product…

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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: (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: Determine the values of y such that D(9,3,-2), E(3,4,7) and F(21,y,-20) are collinear

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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 (X, t) be a topological space and A, B EX, then 1- (X- A)° CX-Ā 2- (X-A)° 2X-Ā 3- (X- A)° = X-A…

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Q: Homworke 1.1 Prove that the Euclidean space R" is a metric space under the n Euclidean metric d(x,…

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Q: Let L: R2 → R2² be the linear operator defined by L 2

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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: Thm. Let V be an inner product Space and ждек then | 1 ≤ 11all. 1171/ further equility is hold iff…

A: Using the properties of inner product And  aā=|a|2 , for any complex scalar  ||cx||=|c| ||x||, for…

Q: r', where i =

A: As per rule, allowed to answer first question and post the remaining in the next submisison.

Q: Let (x, e) and (y, d) be metric space with e(x, y) = { 1 and let fr→y be a function. Is f…

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Q: d1(x,y) = e*-y, d2(x,y)= |2x – yl and d3(x,y) = /]x – yl %3D %3D O Both d1 and d2 are metrics on R.…

A: We have,d1x,y=ex-yd2x,y=2x-yd3(x,y)=x-yWe will check whether they satisfy the properties of metrics…

Q: Give the definition of a separable metric space (S, d). Show that the metric space (R, d), where…

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Q: c spaces. Show that X x Y and Y

A: The sup-metric d1 on X×Y is defined as d1x1, y1, x2, y2=max dx1, x2, ey1, y2. Similarly, the…

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) =…

Q: If (X, d) is any metric space, show that another metric on X is defined by d(ry) d(r, y) 1+ d(x, y)…

A: The given problem is to show that X is bounded in new metric d(x,y)  Already given (X,d) is the…

Q: Prove that the d1 (“Manhattan”) metric on R2 is really a metric

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Q: If X =R and d(x,y)= 1, then X not metric semi metric metric space

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Q: 0. Show that C[-1,1] c C[0,1]. C[0,1]).

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