H.W.2 Exercise Prove that: 1. (IR, J- |) is a complete metric space. 2. (Q. |-1) is not 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: Is the euclidean space R³ is a complete metric space

A: We have to check whether euclidean space R³ is a complete metric space or not .

Q: If the dual X' of a normed linear space X is fininte dimensional, then X is finite dimensional

A: This is the question of Functional Analysis.

Q: Lemma 2.56 Let (X,T) be a topological space, (M, d) be a complete metric space and BC(X,M) := {f €…

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Q: Let (R. d) be discrete metric space, then Ris separ able O True O False

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Q: Prove C(x) with the supremum norm is a metric space by showing the supremum norm is a metric

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Q: Prove that if the space X' of a normed linear space X is separable (as a metric space) then X itself…

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

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

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Q: Let (X,d) be metric space and X is unbounded, then d (z. y) . y EX, is 1+d(r,y) di de fined as dy =…

A: Metric space

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

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Q: The Cartesian product X = X1 x X2 of two metric spaces (X1, d1) and (X2, d2) can be made into metric…

A: 4) To define a new metric on X which is different from the metrics mentioned in (1), (2), and (3).…

Q: メ=D¢ c omplex number %3D Doyou CGid) is metric space of not ??

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Q: (b) Show that every compact set of T₂-space is closed set but the convers need not to true. (c) Let…

A: T2-space: A topological space X is said to be T2-space for given x, y∈X, there exists open sets U…

Q: Lemma 2.56 Let (X,T) be a topological space, (M, d) be a complete metric space and BC(X, M) := {f e…

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Q: A completely regular space is also T1-space 1 True 2 False

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Q: Let (R,d) be diserete metric space then R is not compact . True or false.??

A: Let (R,d) be diserete metric space then R is not compact . True or false.??

Q: Let (X, T) be a topological space and A,B CX- ConstYuct the general Yelation between Fr (IntlA) and…

A: For any subset B of a topological space X , Frontier or boundary of B is defined as  Fr(B) = CL(B) -…

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

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Q: Let (R, d) be the metric space with d(x,y) = |x – yl. Then the open ball B(1,2) is None of these O…

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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 = R and def ine ||.|| on X as ||x|| = x² then X is normed linear space True False

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Q: Lemma 2.56 Let (X,T) be a topological space, (M, d) be a complete metric space and ВС (Х, М) :%3D…

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Q: . Prove or disprove every metric space is normed space,

A: A normed vector space or normed space is a vector space over the real or complex numbers, on which a…

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 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: Show that ℓ^1 is a normed linear space.

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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: Show that R is a bianch space.

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Q: Let (X, d) be metric and PXIR space, dosy) I+ dexy) a de fined by qusN pusy) = ニ show that p is…

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Q: Q # 3(b) Let (G, H) be a disconnection of a topological space (X, 7), ant let AC X. If A is…

A: Before going to this proof, we prove the below statement which in turn proves the above result.…

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

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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: Let (X,d) be a metric space with the added condition that for any three points x,y,z in X, d(x,y) 0…

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Q: Find all limit points of a set A = {(2+,2 – 1)|n e N} on the Euclidean plane with the usual metric.…

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Q: Q7: Prove that (R", ||-|| ) is complete metric space.

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Q: Let X=R and T = {]a, +oo[/a R}U{o,R}, then (X, 7) is a regular space. True False

A: This is a problem of topology.

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

A: ANSWER :

Q: Show that it A is subset of X any with the discre te metric then %3D

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Q: Show that the set of all real numbers constitutes an incomplete metric space if we choose d(x, y) =…

A: Since you have asked multiple question, we will solve the first question for you. If you want any…

Q: Let X,d) be a metrie space. Define d': Xx X >R by: d'Cxoy) Show that (X.d') is min (dl8), 1). a…

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Q: Q7: Prove that ( IR", II-||) is complete metric space.

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Q: If X is a metric space and Y C X, then Y is a metric space.

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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: The set R2 is a vector space over R. How Prove it ?

A: We need to check each and every axiom of a vector space to know that it is in fact a vector…

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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Q: Let (X, || · ||) be a normed linear space. Prove that (a) B1(0) = B1(0).

A: Since you have asked multiple question, we will solve the first question for you. If you want any…