Q2: In Euclidean metric space (R2, 1. 1), is the set A = {(x, y): 2x² + y² = 1} compact? giving the reason. closed bounded Q3: Let S be a closed subset of a compact metric space (M, d). Prove that S is compact in M.

Q: Ametric space (x,d) is complete if and only ifx has the cantor intersection property

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

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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: every metric space is normed space? give an example to show tha

A: We have to show that every metric  space  may not be a  normed  space.

Q: Let R be the set of real numbers and d: R x R → R be defined as follows. d(x, y) = 5|x10 - y1º|.…

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Q: Exercise 3. In this exercise (X, d) is a metric space. .} 1) Assume that X is compact. For every r>…

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Q: 5. Let (X, d) be a metric space. i) Show that the open ball B, (ro) = {x € X : d (T, xo) < r} is an…

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

Q: If X is vector space and (X,d) is discrete metric space, then X can induce an inner product space…

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Q: Q7) (a) Let X be a Banach space and R the usual normed space. Show that ||(z, a)|| = ||||+|al is a…

A: (a) Given a Banach space  and  the usual normed space, we want to show that the norm  is complete on…

Q: Suppose (X, d) is a metric space and E a non-empty subset of X. Prove that E is disconnected if and…

A: Use definition of disconnectedness mentioned below.

Q: Let (X,d) be any metric space, then the only open sets and closed sets are and X O True O False

A: Given statement is false because if we take descrete metric space

Q: ctors - Z. S. Madhi

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Q: Let f be a continuous mapping of a metric space X into metric space Y,g be a continuous mapping of…

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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: 3. Let (X, d) be metric space, prove that for any P1, P2, P3,..., Pn E X, we have d(p1, Pn) ≤d(p1,…

A: We prove the result by mathematical induction We know the triangle inequality of metric space If x,…

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

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Q: let (2, m) be a meesurable space. then 1f f9: x LJare measural ble function, then -> ) f43 is mea…

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Q: (e) Show that a compact metric space can be covered by finitely many balls of any given radius. (f)…

A: A compact space has the property that every open cover has a finite subcover. In the context of…

Q: / Let (X₁T) and (Y₁J) be top. Spaces and fix-x be a surjective and continuous map, then y is Compact…

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Q: Theorem 2.22. Let A and B be subsets of a topological space X. Then: (1) A C B implies Ā c B. (2)…

A: Theorem 2.2 (i) Let A and B subsets of a topological space X such that A⊂B i.e. A is contained in B.…

Q: If A is a subset of a metric space X, then show that A = (x: each neighbourhood of x intersects A}

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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: 5. In any metric space (M, D), prove that D(a1, an) ≤ D(a₁, a2) + D(a2, a3) + + D(an-1, an), for a1,…

A: Given M,D is a metric space.   The triangle inequality holds in a metric space, which means for…

Q: Exercise. Prove that in a metric space (X,p), a set G CX is open if and only if it is a union of…

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

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Q: Let (X, d) be a metric space. The diameter of a non-empty subset M of X is defined as 8(M)= sup d(x,…

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Q: 3: Prove that any reflexive normed space is complete.

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Q: be a n-dimensional normed space, then prove that its dual space is also n-dimensional.

A: Let X be a normed linear space. Let X* be its dual space with the usual dual norm ||T|| = sup{…

Q: Prove that (F,|.|) is Banach space where FE{C, R}

A: Definition required:  Let X be a metric space and let xn be a sequence of points in X. (a) Cauchy…