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

Q: 2. Let (S, d) be a discrete metric space and p E S. Identify S i(P), S1 (p), S2(p).

A: We shall provide answers for first question only as per the answering guidelines. For other…

Q: (6) Let (X,M.) be a measure space, and {Ej}cM Prove that : (I) If E,SE,S…. ,then j (II) If E,2E2...…

A: The given problem is related with measure theory. Given that X , M , μ is a measure space and Ej1∞ ⊆…

Q: 69. Let F(x, y) = (x,0). Prove that if C is any path from (a, b) to (c, d), then F. dr (o2

A: Given: F(x, y) = <x, 0>. C is any path from (a, b) to (c, d).

Q: 7. Let x E R and A be a nonempty subset of R. If x is not a limit point of A then x is an isolated…

A:

Q: If u (E) =0, then E is measurable.

A:

Q: Show that the Stein operator for Poisson P(Y = k) = is Tf(r) = Af(k + 1) – kf(k).

A: Given information: The Stein operator for Poisson P(Y = k) = λk/k! is Tµf(x) = λf(k + 1) – kf(k).

Q: Prove that if h E L'(R) and [ h= 0 for all s E R, then h = 0.

A: Use the Leibniz integral rule: For ∫axbx fx, tdt Where -∞<ax, bx<∞ We have, ddx∫axbx fx,…

Q: If sup(S) exists, then it is unique ylgn ihi

A: The solution is given as

Q: is absolutely and uniformly conyergent for all vlues of x an t in the circle 121 <B-¹

A:

Q: 25 If A E M, is symmetric, prove that AT is symmetric.

A: Symmetric matrix

Q: 2. Let f be defined on an open interval (a, b) and assume that for some e e (a, b) we have f'(c) > 0…

A:

Q: Let S= (-2, 3]U(4,5). The largest possible value of E SU such that Select one:

A: Given: S = {-2, 3} ∪{4, 5}   To find:  The largest possible value of ε &gt; 0

Q: Blank 1 is movement of a side of a preimage parallel to that side with respect to one other fixed…

A: We have to fill the blanks.

Q: If the Wronskian W of f and g is 3e^(4t), and f(t) = e^(2t), find g(t)

A: Wronskian W of f and g is given by:

Q: If E, and E2 are measurable then (E,UE,) is not measurable .

A:

Q: of points of S. 6.30 Show that a set consisting of a finite number of points in E is bounded. Gnita…

A: Given, a set consisting of a finite number of points in E.

Q: If E, and E2 are measurable then (E,UE) is not measurable .

A:

Q: If S is a subset of T , then u* (T) su* (S).

A:

Q: If x is a lower bound for S in R then x must be unique..

A: If x is a lower bound for S in R then x must be unique.

Q: Let c E R and suppose that an → a. Show that can → ca.

A:

Q: Let E Os 0s n, 0s zs 1, 8Vzs rS 8. Compute

A:

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: Suppose that f:[a,b] R is bounded, P is parition of [a,b] and Q is refinement ofP, then: L(f;P) <…

A: Answer

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: If E, and E2 are measurable , then (E,UE,) is not measurable . صواب İhi

A: Here we have given E1 and E2 are measurable sets

Q: IF f:x-у and vCy proof fy/v)=x/f"(v) and proof f is continous iff f'(E) is closed set in x for every…

A:

Q: 1. Let TE L(V) and suppose v is in the range of T and w is in the null space of T". Show explicitly…

A: To solve we will use the definition of adjoint operator.

Q: Show that E is a measurable set if and only if for every E > 0 there is a closed set F and an open…

A:

Q: For G = S, state Question 11. necessary and sufficient conditions on T , a subset of S = {1, 2, . .…

A:

Q: Let x, y, z E Z be such that both odd, and z is even. Prov

A: Introduction: Even and odd numbers are distinguished by the following characteristics: Even Number:…

Q: Which of these best describes T?

A:

Q: Suppose a E Z. Prove that for any n E N, 5|2"a implies 5|a.

A:

Q: Let X be a nonempty set and fix x E X. Define (1, rE A (A) = 10, 4 A

A:

Q: Question 11. of S = {1,2, ..., n}, for G(T) = GT. For G = S, state necessary and sufficient…

A: We need to state necessary and sufficient conditions on T, a subset of S={1,2,...,n} for G(T)=GT

Q: If g(a) = a°(exin²a+eoa²a), then gʻ(a) is: atcos.

A:

Q: 4. Let f : R" → R", x+ Ax + b be an affine function, where A E R"xn,beR". (a) Show that if X CR" is…

A: Let f: ℝn→ℝm be an affined map defined as x↦Ax+b, where A∈ℝm×n and b∈ℝm. Then it follows: fx=Ax+b…

Q: For any k = 1, 2, .... n, find E T₁+T2+ T₁+T2+ ... + Tk + In,

A:

Q: To which of the following does the Well-Ordering Principle directly apply? x(x + 1) x E Z 2 z*} A. O…

A:

Q: If S is a subset of T, then u*(T) su* (S)

A:

Q: If S. F.dr is independent of the path C joining any two points, show that there exists a function O…

A:

Q: If m` E = 0, then E is measurable.

A:

Q: Given: mAB = x Prove: mz1 = 180° - x A D Exercises 25, 26

A: Given:           m AB⏜ = x   To prove:              m&lt;1 = 1800 - x

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: Let fELoi Show that i for any E>o, the set 3- S has a positive measure -

A: For f∈L∞ , f∞=infM&gt;0:μx:fx&gt;M=0For M&gt;0, defining the set AfM=X:1 fx&gt;M⇒For f∈L∞,…

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: 5. Let TE B(H). Show that (ker T)- = ran T.

A: Given: T∈BH and T† is an adjoint of an operator T. To show: ker T⊥=ranT†

Q: 1. If f:X → Y and g:Y → Z are functions and gof is onto. a. Must g be onto? Give a proof or a…

A: (a) Given that  f:X→Yg:Y→Z and g∘f is onto To prove : g is onto Let x∈X, y∈Y , z∈Z As gof is onto…

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

A:

Estimation

A new framework of data analysis where the new statistical methods are used for interpreting the results and analyzing the data is known as estimation in statistics.