For X and Y normed linear spaces, let {Tn} be a sequence in L(X,Y) such that Tn → T in L(X,Y) and let {un} be a sequence in X such that un → u in X. Let ɛ = 1 in the definition of convergence of {Tn} to T in L(X,Y). Show that ||Tn|| < M, Vn e N, where M = sup{||T||, ||T2||, ..., ||TN-1||, 1+ ||T||}, for some N E N.

Q: 1. One can construct Riemann integrable functions on [0,1] that have a dense set of discontinuities…

A:

Q: Σ > lan| < o, (-1, 1] For prove that 2 anx" converges uniformly on n=0 EM:

A: The proof for the convergence on [-1, 1] is shown below.

Q: Let En, n = 1, 2,... be a sequence of measurable subsets of R. (a) If En ≤ En+1 for n = 1, 2, then…

A:

Q: Let (ha)tn)be sequence. that A to hot respe of bounded fumcron of A cenvergesimiformy A to hat respe…

A:

Q: Give an example of a metric space X, d and a sequence {Pn} C X such that {pn} is a Cauchy sequence…

A: Let X=(0,∞) Define d:X×X→ℝ+∪0 by d(x,y)=x-y.i.e. d is a standard metric on X induced from ℝ.Take…

Q: • Exercise 2 Assuming that i) the sequence of functions {f„(x), x € [0, 2)} converges to f(x) in…

A: We will use the basic knowledge of analysis to answer this question properly, in detail. Here, I am…

Q: Let fn(x) = exp(-2")+1, for all r E [0, 1], for all n E N. 1. (fa)n converges pointwise to f: [0,…

A:

Q: Suppose that ak ≥ 0 and that ak^1/3→ a as k → ∞.Prove that sigma,k from 1 to ∞,akx^k converges…

A:

Q: Show that a linear map T:X-→ Y between normed spaces is continuous if and only if {Tx,} is a Cauchy…

A:

Q: # ∙1.) (Cauchy form f defined f(n) exists (i) For each (ii) for of some. on and Taylor Thm) (a, b)…

A: 1) Suppose that  is  defined  on     Also  given that    exists  and  continuous  on  c)  We  have…

Q: (c) For an orthonormal sequence {u} in a Hilbert space H, prove the equivalence of the conditions…

A: Given: un is orthonormal sequence in a Hilbert space H. (i) x⊥un for all n implies x=0. (ii) x=∑x,…

Q: In the nth row of Pascal's triangle, n is greater than or equal to 3, show that C(n,0) ...> C(n,n-1)…

A:

Q: converges weakly to x in a normed linear sp

A: Given that, xn converges weekly to x in a normed linear space X We have to show that Every…

Q: 10 12 14 10 12 14 (b) Calculate dB at t = 2. dt 64 dollars per month (c) Explain what the number you…

A:

Q: 19) Prove or find a counter example to the following statements: a) f is continuous on (a,b) implies…

A: Step 1: Explanation for question

Q: (a) In a metric space (M,d), prove that the sequence (b) = (b, b,b,...) is converge to the point b.…

A:

Q: kpression for "12 times x."

A: Times means multiplication. Sometimes it is denoted by dot.

Q: Let X; ~ X"(1) i= 1,2, . - ·, n and assume they all uidependent are Y = Ź Xi %3D しニ」 ان منتتفينتمنل…

A: Given, Xi~χ12 ,i=1,2,...,n a. Y=∑Xii=1n Therefore, X1+X2+.....+Xn~χn2 above is possible using…

Q: Let f: [a, b] → R be a bounded function. Then f [a, b] → R is Riemann integrable if and only if…

A: be a bounded function. Then we need to show the following statements:" is Riemann Integrable, i.e.…

Q: a) Let fn: [a, b] → R be a sequence of integrable functions, converging uniformly to f: [a, b] → R.…

A: As per our guidelines we are supposed to answer only one question. For the solution of the first…

Q: 3. Consider the usual exponential function f(x) = e. (a) Without using a calculator, use linear…

A:

Q: t fn(x) = sin^n(x) for all x [0,pi]. Prove that fn' is not uniformly convergent on [0,pi].…

A: We can prove it by showing that function is not pointwisw convergent.

Q: For X and Y normed linear spaces, let {T} be a sequence in L(X, Y) such that Tn → T in L(X, Y) and…

A: For X and Y normed linear space  Let { Tn}  be a sequence in L(X, Y) such that Tn → T in L(X, Y)…

Q: Suppose Z ~ N(0,1), find a such that P[–a < Z < a] = 0.75

A:

Q: If the ratio of boys to girls in tennis camp is 3:5 if there are 64 campers, how many are girls 

A: We find out how many girls as follows:  • We first need to determine the total number of parts in…

Q: a) Let (h,). (t) be sequences of hounded functions on A that converges uniformly on A to h,t…

A:

Q: Let (x,d) be a metric space. Let {xn} be a sequence in x: When do you say (x,d) is compact? f)…

A:

Q: If v = {[(xn)]; (xn) is a cauchy sequence in V } Show that (ỹ, ||. || ) is a complete normed space…

A: The question states: "If v = {[(xn)]; (xn) is a Cauchy sequence in V}, show that (~, . ||) is a…

Q: Theorem 7.10. (Classification Theorem) Let u, v E R with 0 < u < v. Consider the test sequences…

A: Let u,v∈ℝ with 0&lt;u&lt;v Given test sequences are 1,nu,nv,2n,n!,nn To show 1=Onu and nu≠O1…

Q: 11. Prove that if f is continuous on (a, b] and lim,a* f(x) exists, then fis uniformly continuous on…

A: The theorem is done .

Q: et G be a o-algebra and DCT is Dominated Convergence Theorem If |X <Y E L' and X, 4 X, then E(XG) $…

A:

Q: For X and Y normed linear spaces, let {Tn} be a sequence in L(X, Y) such that Tỵ →T in L(X, Y) and…

A:

Q: 5. Let h, : [0, o∞) → R be given by h„(x) and let h : [0, ∞) → R be given by I +n %3D h(x) = = x.…

A: Answer

Q: Consider the function Sa(x) = nz" (1- z) for re (0,1]. ) Find f(z) the point wise limit of fa(z) on…

A: We have fn(x)=nxn(1-x)      for x∈[0,1]a) f(x)=limn→∞fn(x)=limn→∞nxn(1-x) =0     for x∈[0,1]b)  we…

Q: The Riemann sum, with the reference function (ad f (a) over the interval- 2, 4, using 2 subintervals…

A: Consider the provided question,

Q: let Chn),(t) be dearuences of bounded Inal lenvegas uni formly repectively Show functrons. on A Lo…

A: Given - Let hn, tn be sequences of bounded functions on A that converges uniformly on A to h, t…

Q: Let (fa). be a sequence of functions f : [0, 1] Rdefined by fa(z) = S2n²z +1 if z<! %3D Then 1. S.)…

A:

Q: À differentiable function f: (a, b) → R is uniformly differentiable on a, b if for every E>0 there…

A:

Q: A function is said to be Hoelder a continuous on [a, b] iff |f(x) – f(x')| 0. Show that if 0 < a <1…

A:

Q: MAΤΗ Μ413 DANIELE ROSSO HW 12 (1) Prove that the function f(r) = is not uniformly contimuous on (0,…

A:

Q: 1. Construct a continuous function f : X → Y between two metric spaces with the condition that there…

A: 1. Construct a continuous function  between two metric spaces with the condition that there exists a…