If X is the subspace of `∞ consisting of all sequences of zeros and ones, what is the induced metric on X ?

Q: Let τ and τ' be two topologies of a set X. Is the family τ∪τ' formed by the open elements of both…

A: Union of two topologies is not necessary a topology.

Q: C^k is the set of all functions that are continuous and have first k derivatives that are…

A: Ck=set of all functions that are continuous and have first k derivatives that are continuous To show…

Q: Give an example of an infinite collection of nested open sets in RR under the usual metric on RR…

A: We given an example of infinite collection of nested open sets in R such that their intersection is…

Q: s paces Xand y, then both T and Ta boanded op erators are

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Q: Show that the Lie bracket defined on L/I is bilinear and satisfies the axioms (L1) and (L2) (Algebra…

A: To show that the Lie bracket defined on is bilinear and satisfies the axioms (L1) and (L2) of the…

Q: f If Xo's Compact, x=y Continuous bijective L Topelagied space Then f is homeomorphism

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Q: Let V, W be finite-dimensional inner product spaces and T € L(V, W). Prove: (a) If T is injective,…

A: Injective Function:  A function is said to be injective if each element of the codomain is mapped to…

Q: Let X be a locally compact Hausdorf f space

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Q: There are several norm sok prac tical importante on the vector space of ordlered s n-tuplesLnotably…

A: A norm is a real-valued function defined on the vector space that is commonly denoted {\displaystyle…

Q: [} [2] » 21. Let u = and v = Show that is in k Span {u, v} for all h and k.

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Q: Q3/(a) Let d be a trivial metric on a set of X, find the induced topology on X by a metric d. (b)…

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Q: Let ECR. A point a E R is called a cluster point of E if En(a-r, a+r) contains infinitely many…

A: Given: E is bounded infinite subset of ℝ. To prove: E has at least one cluster point.

Q: Consider the Banach space C[0,1] of continuous functions on the interval [0,1] equipped with the…

A: The Banach space C[0,1] is the space of all continuous functions defined on the closed interval…

Q: Give the set of limit points A0 of a singleton A = {(5, 2)} on the plane R 2 with the discrete…

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Q: Show that finite dimensional inner product space is Hilbert space

A: We are to show that finite dimensional inner product space is Hilbert space.

Q: normed lineer spare X is finite demisional Linear transformation. X is bounded om If them every

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Q: Prove that the Peano Curve is continuous and surjective mapping [0,1] onto the unit square

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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: 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: Give an example of a closed subset of R that is not sequentially compact -Give an example of a…

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Q: Let S = { (0,0), (0,3), (1,0), (1,2), (2,0), (3,2) } Find S', the transitive closure of S

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Q: (4) Every T4-space is: (a) T2-space (b) T3-space (c) To-space (d) All of them.

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Q: Let R` be the set of real numbers with the lower limit topology (the collection of the intervals [a,…

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Q: Exercise 8. Show that a function F: R" → R" preserves the norm || · || if and only if it preserves…

A: We have the function F:ℝn→ℝn that preserves the norm · This implies that this function satisfies all…

Q: With the help of exomples Show thaat on finite dimensional are equivalent. any two norms Linear…

A: Given: Any two norms on finite dimensional linear space are equivalent.

Q: 9. For each n E IN, let Xn be the set of polynomials of degree equal to n, and let Pn = U_0 Xk. True…

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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: Jtind the absolute maximun Gnd minimum ualue of He qiun funch on Interval on the Specified f6) =…

A: Given function is fx=2x3+3x2-12x-7 on -3,0 find the absolute maximum and minimum value of the…

Q: metoix space and d[211?(₂) for all ₁2 ER 2,12₂ 17. (*/ 1²(₂) d is matrie X. let (Xid) be d' XXR…

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Q: lidear Prove Shat os ever de is a non toe thede Functional actinp non- zero normed space non tec…

A: This proof follows as a consequence of Hahn-Banach Theorem for Normed Spaces. Hahn-Banach Theorem…

Q: Prove that Mobius transformation can have at most two fixed points

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Q: is an Let V(F) be an inner product space. If 0 + o e V then prove that orthonormal set.

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Q: 1. Let X be a topological space. Suppose X contains an infinite, closed, discrete subspace. Show…

A: Given topological space X such that X contains an infinite closed discrete subspace. We now show…

Q: Show that ([0,1] spack по matric matriz, is complete uniform norm

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Q: Prove that the Joukowski map 1 p(2) : (z + maps ONTO T - (-1, 1)

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Q: Let (M, d) be a metric space and K ⊂ M. The following statements are equivalent: (i) K is relatively…

A: Note: We’ll answer the first question since the exact one wasn’t specified. Please submit a new…

Q: Exercise 1. Give examples of subsets A, B in a topological space X (for example X = R or X = R²)…

A: Disconnected sets

Q: If l is a line through (0,0), the point (0,0) divides the line l into two rays. Also, a…

A: To determine for which rays starting at the origin ez is bounded, we need to analyze the behavior of…

Q: If them every normed lineers spare X is finite demisional Linear transformation. X is bounded a

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Q: 25. Find Uge B and Be B for each collection B. BE 4 (a) + *+{[11] N}*(0) + {[L1 =): N} N}₁ B+ :n (c)…

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