Show complete proof/solution. Exercise 6| Show complete proof and solution.In R" wheren E N, letB,(x) = {y : ||x – y|| < r} andT = {E C R" : (Vx E E), (3r > 0) such that B,(x) C E}Show using the definition of topology that T is a topology on R".

Answered: Image /qna-images/answer/0e541d47-afd1-4098-91ae-3413089e4104.jpg

Answered: Exercise 6| Show complete proof and…

Elementary Linear Algebra (MindTap Course List)

8th Edition

ISBN:9781305658004

Author:Ron Larson

Publisher:Ron Larson

Chapter4: Vector Spaces

Section4.2: Vector Spaces

Problem 38E: Determine whether the set R2 with the operations (x1,y1)+(x2,y2)=(x1x2,y1y2) and c(x1,y1)=(cx1,cy1)...

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Determine whether the set R2 with the operations (x1,y1)+(x2,y2)=(x1x2,y1y2) and c(x1,y1)=(cx1,cy1) is a vector space. If it is, verify each vector space axiom; if it is not, state all vector space axioms that fail.
Q1:A) Let X = R and r be the usual topology on R.If Y-[4,6). 1) Determine the open ,elosed ,open and closed for the following sets: (4,5), (4}. (5,6]. 2) Find 31
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Lemma 2.56 Let (X,T) be a topological space, (M, d) be a complete metric space and BC(X, M) := {f e C(X, M); f[X] is bounded } do(f, 9) := sup d(f(x), g(x)) reX (f,9€ BC(X, М)). Then (BC(X, M), d) is a complete metric space. 24 Proof: excersises.
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Exercise 6| Show complete proof and solution. In R" wheren E N, let B,(x) = {y : ||x – y|| < r} and T = {EC R" : (Vx E E), (3r > 0) such that B,(x) C E} Show using the definition of topology that T is a topology on R".