3. Prove that the image of a measurable set E under the translation 4 is measurable and m(E+a) =mCE).

Q: 3.2 Prove that in any metric space (S, d) every closed ball S,xo is a closed set.

A: Let (S, d) be a metric space. Consider a closed ball Srx0 with both center x0∈S and radius r>0…

Q: Suppose that E1 and E2 are measurable sets. Prove that m(E1 U E2) + m(E n E2) = m(E1) + m(E2).

A: We have to prove that the given statement.

Q: 2.4 (Ross, 2003, page 593) Show that the relationship $ (x) = x;¢(1;, x) + (1 – x;)ø(0¡, x) is true…

A:

Q: 6. Compute the norms for (a)||f()||, (b) ||F(x)||2, (c)||f(x)||, where f(x) = -V1-r", defined on [0,…

A: Given

Q: Find an example of a bounded convex set S in R2 such that its profile P is nonempty but conv P ≠ S.

A:

Q: Let (X, E, µ) be a measure space. Justify through an example that the hypothesis µ(X) < o in…

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Q: Let XC Rn convex. Prove that x is an extreme point of X if and only if the set Cis convex, where X =…

A: Let X⊂ℝn be convex . To prove: x is an extreme point of X if and only if the set C is convex, where…

Q: Theorem 6.2. Let C be a compact subset of Rstd· Then C has a maximum point, that is, there is a…

A: We start the proof by assuming there are no such maximum element in C.

Q: Consider X = C[0, 1] with the norm = max |x(1)|, IE[0,1] and let T: X - X be the mapping given by…

A: note : As per our company guidelines we are supposed to answer ?️only the first question. Kindly…

Q: In a normed linear space X, prove that the closed ball, ū (x, r) is a convex set for any хE X, and…

A:

Q: Determine whether is a linearly independent set of vec- 3 11 tors in R3.

A: A set of vectors v1,v2,v3,…,vk in a vector space is said to be linearly independent if…

Q: A and B are bounded subsets of R, then prove that UB and AnB are bounded.

A:

Q: 3- A discrete topological space (X,T) is: a-To-space, b-T₁-space, c- T₂-space, d- No one.

A:

Q: Give an example of a set X and topologies T1 and T2 on X such that T1 union T2 is not a topology on…

A:

Q: If S, T are nonempty bounded subsets of R with S ⊆ T, then inf T ≤ inf S ≤ sup S ≤ sup T.

A: Since S ⊆ T it follows taht inf T ≤ t, ∀ t ∈ T, then  inf T ≤s , ∀ s ∈S:Therefore inf T is  a lower…

Q: m) Show that if T;,T; and Ty are topologies on a set X.then r, nnt, is a topology on X.

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

Q: Prove that the set {x e R : 10/x - x > 0} is bounded. Prove that the set {x E R : x² – 25x > 0} is…

A: To prove that the set x∈R:10x−x&gt;0 is bounded. To prove that the set x∈R:x2−25x&gt;0 is unbounded.

Q: Let (X, T) be a topological space and A C X. Prove that A =A.

A: PROOF :

Q: Show that X and Y are independent iff E(f o X – go Y)² E(f o X – E(f o X))² for all Borel-measurable…

A:

Q: Theorem 2.30. Let A be a subset of the topological space X, and let p be a point in X. If the set…

A: Given that A is a subset of the topological space X and p is a point in X. Consider a sequence…

Q: Let x E RP be a cluster point of a set E and that x ¢E. Prove that x is a boundary point of E.

A:

Q: Let (X, M, u) be a measure space wE

A: Given, (X, Μ, μ) be a measure space where μ is a probability measure. Assume thatf, g : X→ℝ are…

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

A:

Q: 2. Let S,T C R" with S being a proper subset of T. If span(S) = span(T), then T is linearly…

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Q: Let S be a nonempty bounded subset ofR. If for every e > 0, there exists an x E S such that x > b -…

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Q: Let E and F be measurable sets each having finite measure. Prove that m(EAF) = 0 if and only if m(E)…

A: To prove: The statement "Let E and F be measurable sets each having finite measure. Then, mE∆F=0 if…

Q: 1. Let T and S be topologies in X. Show that SnT is also a topology in X.

A: 1.

Q: 4. Let (X, d) be a metric space, r E X, e > 0, and E = {y E X : d(x, y) < e}. Show that E is closed.

A: Let (X,d) be a metric space, with x∈X , ε&gt;0, and E={y∈X:d(x,y)≤ε} To show: E is closed. That is…

Q: Let T and T 'be two topologies of a set X. Is the family T U T´formed by the openings common to both…

A:

Q: . Show that if S is a subbasis for topologies T1 and T2 on a space X, then T1 = T2.

A:

Q: .2 Prove that in any metric space (S, d) every closed ball S,[xo] is a closed set.

A:

Q: Show that; is a linearly independent set in R3.

A: S=121,011,201 Let a,b and c be the scalar in R3. Then write the set in term of linear span.…

Q: 1- Suppose that ,p) is a meric space, an that E CX. Then the interior E is open (m X) and the…

A: The given question is related to metric spaces. The solution is explained below.

Q: 11 Prove that if A CR and |A| > 0, then there exists a subset of A that is not Lebesgue measurable.

A: As per the policy, we are solving a first question. Please repost the question and specify which…

Q: of E. 4. Let E CR be a bounded set. Prove sup E e Ë.

A: (4) Let E be a bounded subset of R  and let u be the sup E.  Then two cases arise-  Either u belongs…

Q: 2- In a metric space (X, d), A is the intersection of all closed subsets of X which contains A.

A:

Q: a) Suppose that E is a set. If there exists a function f from E onto N, then E is at most countable.

A: (a) False f:E → INLet E= IR  Thenf: IR → INdefened as f(x) =  1  , ∨ x ∈ IRi.e There exist a…

Q: The set E={xER:xs2x² – 1} with the usual metric is compact.

A:

Q: A topological space (X, t) is a T1-space if and only il for each x e X, the singleton set {x} is…

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Q: Consider the set F=(-infinity, 1) U (9,infinity) as a subset of R with d(x,y)=abs(x-y). Is F closed?

A:

Q: 4.Describe the largest set S on which it is correct to say that f is continuous S (x,y) = In(1–x² –…

A: Given problem:- Describe the largest set S on which it is correct to say that f is continuous C.…

Q: Exercise 4.11 Suppose m is Lebesgue measure and A is a Borel measurable subset of R with m(A) > 0.…

A: Given, m is Lebesgue measure and A is a Boral measurable subset ofℝ with m(A)&gt;0.  To prove that…

Q: Given, we have a finite purely non-atomic measure space (X, F, µ), prove that µ(F) = [0 µ(X)].

A: Given: Here, the given is finite purely non-atomic measure space (X,F,μ). To prove that μ(F)=0, μ(X)

Q: )Š is the smallest closed set in M such that S CS.

A:

Q: 1. Let X be any nonempty set and let p E X, Prove that : E, {G X:p¢ G}U{X} is a topology on X.

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

Q: If e-ispure imaginary W hat Restriction is placed on z?

A:

Q: Let E ⊆ R. Show that E is measurable if and only if χE is measurable

A:

Q: 4.2 Let Fj, be a closed set for k = 1,2, ... ,n in (S, d). Show that U Fk is closed.

A: