Let x1 and x2 be distinct points in the metric space (S, d). Verify that there are open balls S, (x1) and Sp2 (x2) which are disjoint.

Q: If there exists open ball B (a, r) such that B (a, r) NG= Ø then a is not limit point of G True O…

A: TRUE

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

A: We will prove the given statement.

Q: Let (X1, d1) and (X2, d2) be separable metric spaces. Prove that product X1 × X2 with metric…

A: A matric space X is separable if it has a countable dense subset.

Q: Let X = R, und d be the discrete metric, und A = (3,5) Show that if A is compact or not

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

A: Please check the answer in next step

Q: If there exists open ball B (a, r) such that B (a,r) nG = Ø then a is not limit point of G True…

A: TRUE

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…

A:

Q: E be a Euclidean space of dimension (E(1, f)) = dim(Ker (f- id)), then s minimal.

A:

Q: Suppose Show that p: X²> t0,00) defined by d is a metric n X, 2. 3d (x,y) 2+3d (xy) metric an p…

A: Given that d is a metric on X. We need to prove that ρ is a metric.

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…

A:

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: Let A be a non-empty subset of the metric space (X,d). show that |d(x, A)- d(y, A)| < d(x, y)For all…

A:

Q: Let X = R, and Let M = [0, 3] and d be the discrete metric, then M is

A: M=[0,3] is compact ,since it is closed and bounded in real R.

Q: Let X and Y be normed linear spaces, and T E B(X,Y). Prove that 38 > 0 such that 8||x|| < ||T(æ)||,…

A: Consider the given information.

Q: Let (X,d) be a metric space. Prove that the function d' :X x X →R given by d (x, y)= min {1,d(x, y)}…

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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…

A:

Q: Let (X, T ) be a topological space, (M, d) be a complete metric space and BC(X, M) := {f ∈ C(X, M);…

A: Given as X,T be a topological space,M,d be a complete metric space and BCX,M=f∈CX,M;fX is bounded…

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: Given any Hermitian space E with › E, if (f(x), x) = 0 for all x € E, th I

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

A:

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

A:

Q: Let (X, d) be a metric space and B = S(xo,r) be the open ball with centre at xo E X and radius r> 0.…

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

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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: For every ,x, y E R, let p(x, y) =| x² – y² |. Is p a metric -

A:

Q: Prove by sequences that the open balls in R^n with the usual (Euclidean) metric are not compact

A: In this question, we have to show that open balls in ℝn with the usual metric are not compact by the…

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: Define on R, the metric d defined by d(x, y) = |1+1리 1+ |y| Then (R, d) is not complete. True False

A: Given

Q: Let, T: R' → R';T(x, y,z) = (x+ y+ z,2y + z,2y +3z) Find the dimension of KerT and ImT

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Q: Let A be a non-empty subset of the metric space (X,d). show that a. d(x, A) – d(y, A)| < d(x, y)For…

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Q: Let E be a Euclidean space with inner ratic form. For all u, v E E, we have the C y(u, v)² ≤…

A:

Q: 1. Show that if & is a function from a nonempty compact metric space X to itself such that d(@(x),…

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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: 12. Show that d((x1, Y1), (x2, Y2)) = |x1x2- Y1Y2| is not a metric on X = R %3D %3D

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Q: Let A be a subset of R with the usual metric such that A=A. If there exists r>0 such that d(x,y) <r…

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Q: For normal linear spaces X, Y, prove that BL (X, Y) is complete if Y is complete.

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Q: Let (X, d) be a compact metric space and T: XX be a mapping that satisfies d(Tr,Ty)<d(r,y) for all…

A: What is Metric Space: Metric space is an abstract set in mathematics, particularly topology, with a…

Q: 14. If d is a metric on a vector space X‡ {0} which is obtained from a norm, and å is defined by…

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Q: if x is a compact Metric space and F e C (X₂Y) is equicontinuous, then F i's uniformly…

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Q: An Elliptical norm for R2 is defined by + 4u2, where x = (1,u). (i) Prove that l is a norm for R2…

A: This is a problem of Functional Analysis.

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

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Q: Consider E = C( [o,(]) as normed spaces with normS II f | = sup 1fcxl and l fll,=Sifwildx. %3D Show…

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Q: Let X is a normed linear space and the ball M= {xe X:|| x ||≤1} is compact, then X is finite…

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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…