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||
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Question 3. (4 marks) Let C([a, b]; R) be the space of all continuous functions on...
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...
(Functional analysis) Let C be the space of all functions having Question 4. (3 marks) Let C([0, 1]) be the space of all functions having continuous derivative For each fe C(0,1), set 1/2 1 0 Show that I-1l is a norm of the space of C (0, 1) Question 4. (3 marks) Let C([0, 1]) be the space of all functions having continuous derivative For each fe C(0,1), set 1/2 1 0 Show that I-1l is a norm of the...
- 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...
(1) Let C[a, b] denote the set of functions that are continuous on (a,b). Define a function on pairs of function fig E C[a, b] by dı(,9) = /\f(x) – 9(2)| dx. Prove that (C[a,b], du) is a metric space.
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...
I do not need the two metrics to be proved (that they are a metric). 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....
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....
3. Let f, g : a, b] → R be functions such that f is integrable, g is continuous. and g(x) 〉 0 for all x є a,b]. Since both f, g are bounded, let K 〉 0 be such that |f(x) K and g(x) < K for all x E [a,b (a) Let n > 0 be given. Prove that there is a partition P of [a, b such that for all i 2. (b) Let P be a...
3. Let f, g : [a,b] → R be functions such that f is integrable, g is continuous, and g(x) >0 for all r E [a, b] Since both f,g are bounded, let K >0 be such that lf(z)| K and g(x) K for all x E [a3] (a) Let n > 0 be given. Prove that there is a partition P of [a, b such that U (P. f) _ L(P./) < η and Mi(P4)-mi(P4) < η for all...
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.