Q7: Prove that ( IR", ||·|| ) is complete metric space.

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

A: 3.3 is also known as Hausdorff Property. d(x1, x2) > 0 for every x1 ≠ x2   (metric space…

Q: Let E1, E2, ..., En ⊂ X,d be compact subsets of a metric space X. Just using the definition of…

A: For a metric space we know that the for a compact set every open cover has a finite subcover. So…

Q: Let (R. d) be discrete metric space then R is not complete O True O Faise

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Q: (b) Prove that the set S = {x EQ: √2<x<√3} is closed and bounded in (Q.1-1), but not compact in…

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Q: Let (X, d) be a metric space and let A, B⊆X be such that A is connected, and A∩B ≠ ∅ and A∩ (X − B)…

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Q: Let (X,d) be a metric space and E ⊆ X. Prove that if E is compact, then E is bounded

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Q: A subset I of a metric space R with the usual metric is compact if and if only it is an interval…

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Q: Let (R. d) be discrete metric space, thend can not induces a norm on X O True O Fahe

A: Thanks for the question :)And your upvote will be really appreciable ;)

Q: Q7. Assume that (R,T) is the usual Euclidean topological space. N = {1,2, 3, ....} C R. Determine…

A: There is no interior point of N.

Q: Let (X,d) be a metric space. For x,y in X define e(x,y)=min{1,d(x,y)}. Prove that (X,e) is also a…

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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, d) be a separable metric space. into l∞.

A: This is a problem of functional analysis.

Q: Consider IR as a metric space.with metric dux, y) = 1x-y1. Show that the open interval C0, 1) is an…

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Q: Let (X, d) be a metric space, and p: X→ IR doxy) I+desy) de fined by puM pusy) = %3D show that p is…

A: Consider a set X and a function defined from it to the real numbers as d : X×X→ℝ. Then, X,d is a…

Q: 5. State whether each of the following statements is true or false, in either case substan- tiate/…

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Q: In Euclidean metric space (R, I. I), prove that (1+1)→ 1 as n → ∞o in R.

A: as per our company guideline we are supposed to answer only one qs kindly post remaining qs in next…

Q: 10. If A is 3 x 3 with rank A = 2, show that the dimension of the null space of A is l.

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Q: 12 Let (RTR) is topological Space Such that TR-243 U2GSR=R-G-G is finste} Find buz), Z², Z. z' then…

A: Here given ℝ, Tℝ is a topological space such that Tℝ=ϕ∪G⊂ℝ: ℝ-G=Gc is finite This topology is known…

Q: Q3 Assume that (M,d) is a Compact metric Space. Show that, f: (M,d) → (Y, 2) is Continous bjective…

A: f:(X, d) to (Y, d) is said to be homeomorphism if  1) f is continuous.  2) f is bijective.  3) f…

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

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Q: Prove that: 1. (R, |. |) is a complete metric space. 2. (Q.I-1) is not complete metric space.

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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: If S be a n-dimen Sional normed Space, then prove that its dual space is al so n- dimen Sional .

A: A normed space E is called a Banach space if it is complete, i.e., if every Cauchy sequence is…

Q: Let X be a finite dimensional norm space. Then prove that M = {x € X |||x|| <1} is compact.

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Q: 3. Find the closure and interior of the set A = {(x, y): xy = 1} as a subset of R? (equipped with…

A: Closure and interior

Q: let X be a firite dimensional notm Space Then Prove that is compat.

A: As X is a finite dimensional norm space, this means that, dimX is finite.

Q: Prove that in any metric space (S, d) every closed ball S,[xo] is a closed set.

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Q: A of a metric space (x, d.)

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Q: Let (R, d) be discrete metric space, then R is not separ able O True O False

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Q: (a) Prove that every closed subset S of a compact metric space (M,d) is compact in M.

A: Note: Our guidelines we are supposed to answer only one question. Kindly repost other question as…

Q: 1. Let M, d be a metric space with A C B C M. Suppose that B is totally bounded. Prove that A is…

A: Let (M, d) be a metric space and let A be a nonempty subset of M. The diameter of A is defined as…

Q: Let x be an interior point of a subset S of a metric space (X,d). Show that x must also be a limit…

A: Interior Point: Let A be a subset of metric space X, d. A point x∈A is called an interior point of A…

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

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Q: Q2: In Euclidean metric space (R2, 1. 1), is the set A = {(x, y): 2x² + y² = 1} compact? giving the…

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Q: Use the Completion Theorem below to show that the Completion of the discrete metric space X is…

A: By completion theorem, there exists a complete metric space which has a subspace that is isometric…

Q: H.W.2 Exercise Prove that: 1. (IR, J- |) is a complete metric space. 2. (Q. |-1) is not complete…

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Q: Let (R, d) be the metric space with d(x, y) = /[x – y[. Then the open ball O [-3, 5] ]-3, 5[ ]-1, 3[…

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Q: Q6: (a) In any metric space (S, d), prove that every compact subset T is complete. (b) Prove that…

A: A metric space is a set together with a metric on the set. The metric is a function that defines a…

Q: c) Let (M, d) be a metric space and let S,T≤ M. Prove that SOT = ST. But SOT SOT.

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Q: the usual metric space defined by  d(x,y)= x-y prove the four propertis of metric space

A: Applying four properties to this metric

Q: 5. State whether the following is true or false m) A subset I of a metric space R with the usual…

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Q: Given, we have a finite purely non-atomic measure space (X, F, µ), prove that µ(F) = [0 µ(X)].

A: Given: Here, the given is finite purely non-atomic measure space (X,F,μ). To prove that μ(F)=0, μ(X)

Q: Show the following result: Theorem. Let (X, d) be a metric space, A C X, and let u: A → R be an…

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Q: (b) (i) Give an example of an infinite collection of nested open sets in R under the usual metric on…

A: Here given usual metric on the set of real numbers R

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

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Q: Let E and F be compact sets in a metric space (X,d). Show that E U F is compact using the definition…

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Q: Prove that: 1. (R, J. |) is a complete metric space. 2. (Q, I. |) is not complete metric

A: To prove that R,. is a complete metric space.  Suppose that xn is increasing and bounded, Let…

Q: Show that the real number R with d(x,y)=|x-2y| are not a metric space

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