(Functional analysis) Let C be the space of all functions having
(Functional analysis) Let C be the space of all functions having Question 4. (3 marks) Let C([0, 1]) be the space of al...
Question 3. (4 marks) Let C([a, b]; R) be the space of all continuous functions on [a, b], 0 <a<b with the metric || f – 9|| = maxasaso \f (x) – g(x)]. For each f e C([a, b]; R), define a map F(f) by F(f)(x) = x5 + Vx € (a,b]. (65 – a5) Prove that there is a unique fixed point of F in the space C([a, b]; R); i.e. there is a unique fe C([a,b); R) such...
a) 13 marks Let C0, 1 be the vector space of all continuous, complex-valued func- tions on the closed interval 0, 1. Define = (If(a)2 dx sup (x) xE[0,1 112 and (i) Show the triangle inequality ||f + g||00 || ||0 ||9||00 (ii) Show that for any function f e C[0, 1, the inequality ||f||2 ||£||2 holds. (iii Show that there exists no fixed constant C such that the inequality SIl0Cf2 holds for allfE C[0, 1 (iv) Construct a sequence...
Consider the space V of continuous functions on (0, 1] with the 2-norm 12 J f2 We saw in class that V is an incomplete normed linear space. (a) For a continuous function p on [0, 1], define a linear map Mp: V-V by Mpf-pf. Show that Mp is bounded and calculate its norm. (b) Is A = (Mplp E C(0,1)) a Banach algebra? Note that B(V) is necessarily incomplete, so it is not enough to prove that A is...
(4) Let C[0,1] be the inner produce space of all real-valued, continuous functions on the interval (0,1) with inner product.g) = Sopr)(x) dr. Determine the projection of the vector {m} onto the space spanned by the orthonormal system of vectors given below. {1, 73(2x - 1)}
33. Let C[0, 1] be the space of real-valued continuous functions on [0, 1] with inner product Kf.9) (x)g()d(r). 2 cos 2Tir and g.(r)-v2 sin 2mix for i 1.2,... Show that (1. fi.g. f2 92 Suppose that fi(r) is an orthonormal set.
Both part of the question is True or False. Thank you Problem 1. (ref. Example 3 in the slide) Let X = Y = C[0, 1] (with the norm || ||C[0,1] = sup |u(x)]). For any u € C[0, 1], define T€[0,1] v(t) = u(s)ds. We denote by T the mapping from u to v with D(T) = C[0, 1], i.e., v(t) = Tu(t). Then, the following conditions are true or not? Example 3. We denote by the set of...
Let X be the space of all continuous functions from [0, 1] to [0, 1] equipped with the sup metric. Let Xi be the set of injective and Xs be the set of surjective elements of A and let Xis = Xi ∩ Xs. Prove or disprove: i) Xi is closed, ii) Xs is closed, iii) Xis is closed, iv) X is connected, v) X is compact.
b) (10 pts) Let D(0, oo)) be the vector space of all bounded continuous functions from [0, oo) such that R If(x) dz 00. Give an example of a sequence {fn} of functions in D(0,00)) which (i) converges pointwise for E [0, oo) to the constant function f(z)0 (ii) does not converge to 0, neither with respect to the norm, nor the Hint: it may be helpful to contemplate the phrase "mass escaping to infinity". norm. b) (10 pts) Let...
b) 16 marks Assume that each set Vi, j = 1, 2, ...k, is a compact set in a metric space X. Prove that the (finite union) set V = V1 U V2 U... U Vk is a compact set. c) [7 marks] Let H be a Hilbert space with inner product < x, y > and the induced norm ||2|= << x, x >. (i) Show that ||* + y|l2 + ||* – y|l2 = 2(1|x1|2 + ||4||2) for...
B2. (a) Let I denote the interval 0,1 and let C denote the space of continuous functions I-R. Define dsup(f,g)-sup |f(t)-g(t) and di(f.g)f (t)- g(t)ldt (f,g E C) tEI (i) Prove that dsup is a metric on C (ii) Prove that di is a metric on C. (You may use any standard properties of continuous functions and integrals, provided you make your reasoning clear.) 6 i) Let 1 denote the constant function on I with value 1. Give an explicit...