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.
Let X be the space of all continuous functions from [0, 1] to [0, 1] equipped...
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...
1. Let (X, d) be a metric space, and U, V, W CX subsets of X. (a) (i) Define what it means for U to be open. (ii) Define what it means for V to be closed. (iii) Define what it means for W to be compact. (b) Prove that in a metric space a compact subset is closed.
- Let V be the vector space of continuous functions defined f : [0,1] → R and a : [0, 1] →R a positive continuous function. Let < f, g >a= Soa(x)f(x)g(x)dx. a) Prove that <, >a defines an inner product in V. b) For f,gE V let < f,g >= So f(x)g(x)dx. Prove that {xn} is a Cauchy sequence in the metric defined by <, >a if and only if it a Cauchy sequence in the metric defined by...
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...
12. Let M be the set of continuous functions on R which vanish outside a finite interval (the interval may depend on the function). (a) Show that M is a metric space in the sup norm. (b) Show that M is not complete. Chapter 5. Sequences of Function 210 (C) Show that C.(R), the continuous functions which go to zero et is complete in the sup norm (problem 10 of Section 5.3). (d) Prove that Mis dense in Co(R). T...
Let (X, d) be a compact metric space that has at least two elements. Prove that if y ∈ X then fy(x) = d(x, y) is a continuous function fy : X → R.
Let (X, d) be a compact metric space, and con- sider continuous functions fk : X → R, for k N, and f : X → R. Suppose that, for each the sequence (fe(x))ke N 1s a monotonic sequence which converges to (x). Show that r є X, k)kEN Converges to j uniformly. Let (X, d) be a compact metric space, and con- sider continuous functions fk : X → R, for k N, and f : X → R....
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...
8. More generally, let X be any infinite-dimensional vector space equipped with an inner product ,) in such a way that the induced metric is complete. In particular, there is a norm on X defined by and the metric is given by d(r, y) yl Let A denote the unit ball A x E X < 1} We know that A is closed and bounded essentially from the definitions. Show that A is not compact. (Hint: Construct a sequence xn...
Problem 2. Let C[0, 1] be the set of all continuous functions from [0, 1] to R. For any f, g є Cl0, 11 define - max f(x) - g(z) and di(f,g)-If(x) - g(x)d. a) Prove that for any n 2 1, one can find n points in C[O, 1 such that, in daup metric, the distance between any two points is equai to 1. b) Can one find 100 points in C[0, 1] such that, in di metric, the...