Q Let (X, B(R), µ) be a Borel measure space. Suppose that {A, = [2n-1, 3n+1]} be a family of elements of B(R). Then µ(U 1, 4n-1]) equals n=1 [n+1 n=1

Q: 1. A natural cubic spline S.on [0,2] is defined by (S.(x) S(x) = {x)=1+ 2x-x s,(x) = a+ b(x- 1) +…

A: The cubic spline is defined as: Sx=S0x=1+2x-x3 if 0≤x≤1S1x=a+bx-1+cx-12+dx-13 if 1≤x≤2 From…

Q: Let X be a normed space and aɛ X. Show that there exists an fe X' such that ||a|| = Sup { \f(a)| :…

A: Given that,                X is a normed space and a ∈ X . We have to show that there exists an f ∈…

Q: #2. Consider the function f (x,y) = x +y-xy on the set D which is the closed triangular region with…

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Q: 7. In A ABC,X is the centroid. A A. If CW = 15, find CX and XW. W B B. If BX = 8, find BY and XY. C.…

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Q: The linearization a at a = 0 to √8+7x is A + Br. Compute A and B. A B=

A: To Find: Linearization for fx=8+7x at a=0

Q: H is a Hilbert space, A: H → H is linear ar = for all r, y e H. Use th closed graph theorem to show…

A: For two normed spaces V and W, a linear operator T:V→W is closed if for each xn∈V, xn→x and Txn→y…

Q: Exercise 4.2 Let m be Lebesgue measure and A a Lebesgue mea- surable subset of R with m(A) 0. Show…

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Q: 3;) Find the välie of k isó that. the furnictioin %3D .kx*. .if. x 2 2. is. continuous .cvery iw…

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Q: The main point of this exercise is to use Green’s Theorem to deduce a special case of the change of…

A: We will use the basic knowledge of multivariate functions to answer this question. Note that in your…

Q: 3.20 Composition with an affine function. Show that the following functions f: R" -→R are convex.…

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Q: (H.W) let X=Rbe amatric space show that the folloowing subset of Rare сотраct or not 1 1 1- E,=|1 -…

A: We have to find the subset is compact or not

Q: Let X,,X,be two real-valued features and Y be a Boolean-valued function of the given features such…

A: Solution:   Guileless Bayes classifiers are a variety of request estimations taking into account…

Q: (b) If (X ,7) is a topological space, then the inclusion function i :X →X where i (x ) =x for each x…

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Q: Lemma 4.3 Assume • T(0) = Idx; • T(t +s) = T(t)T(s) (t, s 2 0); • Vx€ X :xH T(t)x is continuous in…

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Q: 3. Let X1,., Xn be iid Uniform(0, 3) where 3 is an unknown parameter. You can following fact without…

A: given that

Q: 3. Suppose g is a step function on [a, b). This means that there exists a partition a = yo < y1 <…

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Q: (H.W) let X=Rbe amatric space show that the folloowing subset of Rare сотраct or not 1 1 1- E,=|1 -…

A: As per our guidelines, I solved first three subparts. Kindly request you to repost the remaining…

Q: 2. Use Wronskian to show that fi = x, f2 = ln x, and f3 C° (R). x In x are linearly independent in…

A: Concept:

Q: Lemma 3.3.3. If X is a normed space and xn, x, Yn, y are vectors in X, then the following statements…

A: As you have asked many questions without specifying, we shall solve first three question only. For…

Q: 3.20 Composition with an affine function. Show that the following functions f: R" -→R are convex.…

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Q: 9. Let be a bounded open subset of C, and y: → a holomorphic function. Prove that if there exists a…

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Q: (b) Given two metric spaces X,p>, and a function f : X→ Y that is uniformly continuous on SC X. If…

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Q: 10. If A is 3 x 3 with rank A = 2, show that the dimension of the null space of A is l.

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Q: Please solve

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Q: 6. Consider the horizonal strip 2 = {z+ iy : -1 < y < 1}. There exists a holomorphic function f on…

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Q: 4. Evaluate the Kullback-Leibler divergence between two Gaussians p(x) = N(xu, E) and %3D q(x) =…

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Q: Let A: C [0,1]→C [0,1]. Prove that operator A is continuous reversible, find A!, 2.11. Ax(1) = x(1)…

A: We have given operator A such that A: C[0,1]-C[0,1].Also we must know that the inverse operator…

Q: 3. (1) Consider A = [0, 1] and B = [0, 1) and injections f(x) = x/3 from A to Band g(x) = x from B…

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Q: The main point of this exercise is to use Green’s Theorem to deduce a special case of the change of…

A: Please note that although it is written in your question post Solve all of them please, but these…

Q: 3.42. 12. Suppose g and Saln= 1, 2, 3, ...) are defined on (0, o), are Riemann-integrable on [t, T]…

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Q: (b) {T } is a sequence of bounded linear maps on a Banach space X and Tx = lim 1x exists for all xE…

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Q: We have seen that the space C-1, 1] with (S. 9) = | s (x) g (x2) dar is an inner product space. Let…

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Q: 38 A system {x;}ien in a normed space X is called a complete system Span {L-1 a;X; : Vn EN, a; E F}…

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Q: Q5. Let (X, B(R), µ) be a Borel measure space. Suppose that {A, = ["+1_ 4n-1 Show that A1 C A2 C…

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Q: 22. (a) Prove that [0, x] = [0, y] for all x, y E Z, x + 0, y # 0. (b) Prove that [ax, ay] = [x, y]…

A: (a) we have to prove that  0, x = 0, y for all x, y∈Z, x≠ 0, y≠0 proof as x, y∈Z, x≠ 0, y≠0 Z is the…

Q: 5) Let X and Y be normed spaces. Suppose T:X → Y be a closed linear operator. Let (xn) be a sequence…

A: We will prove this by using definition

Q: 10. If A is 3 x 3 with rank A 2, show that the dimension of the null space of A is 1.

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Q: (b) Given two metric spaces X,p>, and a function f : X→ Y that is uniformly continuous on SC X. If…

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Q: (a) Suppose ƒ: [a, b] → R is integrable and L(f, P) = U(ƒ, P) for some partition P of [a, b]. What…

A: (a) Given that f:a,b→ℝ is integrable and Lf,P=Uf,P for some partition P of [a, b]. To find: what f…

Q: Theorem 2.10 Suppose X, Y, Xn are integrable. (i) If X = a a.s., then E(X N) = a, a.s. (ii)…

A: Given X, Y, Xn are integrable

Q: Let X = (!,2) and T3} U = (1,2-)inel,2,U+ jult. %3D n+1 show that X is not T - space

A: This is a problem of Topology.

Q: and y-f a. Let =-5 Find the veetoes d %3D %3D

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Q: 2. (a) Let X = C[0, 1]. Define T: X→ X by. Tf(x) = xf(x) Show that (i) T is bounded (ii) Find || T…

A: Given- Let X = C0, 1. Define T : X → X by Tfx = xfx.  To Show- Show that T is bounded. Find T.

Q: 7.2.9 Additivity Theorem Let f := [a,b] → R and let ce (a, b). Then f E R[a, b] if and only if its…

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Q: Theorem [4.1.11] Let A and B be subsets of topological spaces X and Y respectively. Then Int(A × B)…

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Q: Find the Wronskian for the set of functions.{1, x, x2, x3}

A: we have to find the wronskian for the set of functions. {1, x, x2, x3}

Q: Let z = F(x , y) = xy2 + - , where x = u2 + v2 - 4 and y = uv . az Use the chain rule to find at (u…

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Q: Use th properties of linear Space A upon t to prove s a) 0.X - o for all X EX b) a.x=0 =>(a=07v…

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Q: Theorem 7-6. If the variables are uniformly bounded then the condition, lim Bn = 0, n- 00 n2 is…

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