On a previous homework, you proved a Bolzano-Weirstrass theorem for R3 with the metric d((x1, X2, X3), (V1, y2, Y3) = |X1 - yıl + |x2 - Y2l + \x3 - y3]- onclude that, with this metric, a subset of R³ is sequentially compact if and only if it is closed and bounded.

Q: Let M = {0,1} and d be the discrete metric, then d is not complete %3D O True O False

A: Discrete metric space.

Q: A: For each of the following maps T: V→ W, find the spaces V and W and say if they are linear or not…

A: The given map T:V→W such that Tx, y=1, y. We have to check whether the map is linear or not.

Q: Let (X, d) be a metric space, prove that for any x, y, z E X, we have |d(y, z) - d(x, z)| ≤d(x, y).

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Q: (a) Suppose that (M, d) is a metric space. Consider the mapping f: M [0, 1] be given and let di(x,…

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Q: A. Let H be the set of all points (x, y) in ℝ2 such that x2 + xy + 3y2 = 3. Show that H is a closed…

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Q: Prove d(x,y) = 8 if x does not equal y and 0 if x equals y is a metric on R, the real numbers.

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Q: = 0,1] and d be the usual metric, d(x, y) = a -y %3D en d can not induces a norm on X True False

A: the usual metric d(x,y)=|x-y| usually induces the norm.

Q: On a previous homework, you proved a Bolzano-Weirstrass theorem for ℝ3 with the metric d((x1, x2,…

A:

Q: Consider the three basic vectors: (2,1,0,-1),(-1,1,1,1),(2,7,4,1) a. give two vectors that are in…

A: Given the three basic vectors:                           (2,1,0,-1),(-1,1,1,1),and (2,7,4,1) We have…

Q: (a). Define three different metrices d1, d2, d3 and their induced norms on R³. Also prove the…

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Q: Let X be the space of bounded and closed linear segments on the positive part of the real line, so…

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Q: A. Consider R3 with the metric d((X1, X2, X3), (V1, Y2. Y3) = |X1 - Yıl + |x2 - y2l + [X3 - Y3l. i.…

A: This problem is related with metric on ℝ3. Given that                   dx1,x2,x3,y1,y2,y3 = x1-y1+…

Q: E. Let H be the set of all points (x, y) in R² such that x² + 3y² = 12. Show that H is a closed…

A: We have to find H is closed  subset bounded.

Q: Example 3.16. In R with the Euclidean metric, the set [0, 1] is compact. However, note that with the…

A: To Explain: [0,1] is compact in Euclidean metric and not compact under Discrete metric.

Q: 2. Let d1, d2 be two metrics on X + 6. Show that: d(r, y) = d1(x, y) + d2(x, y).

A: I have proved all four conditions of metric for d

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…

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

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Q: Let X= [0, 0 ), and defined d as d(x,y)= e(xy) then verify all four condition of metric space

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

A:

Q: (a) Show that d(m, n) = |m²-n²| 1+ m²-n²| defines a metric on the set of natural numbers N. Hint:…

A: Given That: d(m,n)=|m2-n2|1+|m2-n2|   To Show : a) d  defines a metric on the set of natural number…

Q: (L(x), ll·lla) is normed Space

A: Given: L∞X, ∥·∥∞To prove: L∞X, ∥·∥∞ is a normed space.

Q: r', where i =

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

Q: Let V be a 5 dimensional space, and V: V → V linear. Can we have NullT = RangeT? And if the…

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Q: ) Give definition to the convex set of the space Rn. Also give a definition for when the function f…

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Q: . Show that dim(w,) + dim(W,)– dim(w, n w,) = dim(w, + w, - 1 C

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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: Example 3.16. In R with the Euclidean metric, the set [0, 1] is compact. However, note that with the…

A:

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: The equation, d(t) = 255t + 5654 represents the tuition, d dollars, at a college for the year Find…

A: I have explained everything with in the solution .please go through that.

Q: Problem 2. Let U = {a+bx+ca²+dr³ € P3 (R): a-b+d=0}, 2(i) Find a finite subset S of P3 (R) such that…

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Q: Let (X. d) be a metric space and define p: XxX by ptr.g)= dx.y) Prove that (X,p) is a metric…

A: Introduction: To prove that (X, p) is a metric space, we need to show that the following three…

Q: Consider ℝ3 with the metric d((x1, x2, x3), (y1, y2, y3) = |x1 - y1| + |x2 - y2| + |x3 - y3|. i.…

A: To verify that given distance form d is a metric , we have to check for  Identity Symmetry…

Q: D. Consider R³ with the metric i. Verify that d is really a metric. ii. State and prove the…

A: To verify that given distance form d is a metric , we have to check for IdentitySymmetryTriangular…

Q: d((X1, X2, X3), (V₁, V2, Y3) = |X₁ - Y₁| + |×2 - Y₂ + X3 - Y3l. Conclude that, with this metric, a…

A: Given  is a metric space.To show that a subset  of  is sequentially compact if and only if it is…