Given : B(p,m) = {c C = R+n | pc m} and p > > 0, m 0
Let T : B(p,m) = {c C = R+n | pc m} is uhc at x0 iff , for all open sets V there exists open set U such that x U
T(x) V. T is upper hemicontinuous iff it is upper hemicontinuous at all x.
Let T : B(p,m) = {c C = R+n | pc m} is uhc at x0 iff , for all open sets V such that V T(x0) , such that x U, V T(x)
T is lower hemicontinuous iff it is lower hemicontinuous at all x.
If T : B(p,m) = {c C = R+n | pc m} is correspondence then T is closed graph then it is convex and compact valued.
3. Let B(p, m) ce C RTlpcS m be the budget correspondence for p >>0, m2 0. a. Show this correspon...
15) Show that the fune [6] Let f : (a, b) → R be strictly convex on (a,b). Show that there is 80 cE (a, b) such that f is strictly increasing or stricty decreasing on le,b) some poirnt 15) Show that the fune [6] Let f : (a, b) → R be strictly convex on (a,b). Show that there is 80 cE (a, b) such that f is strictly increasing or stricty decreasing on le,b) some poirnt
12. Let cE C and let bER a. Show that cz + Cz b is the equation of a line in C. b. For which values of b and c is l212+cZ+cz b the equation of a circle in C?
Exercise 5.1.1: Let H = C², M1 = C|0) and M2 = C(0) + 1)) Let [2) = a|0) + B|1) with (al2 + 1B12 = 1. Show that Pr(span{M1, M2}) # Pr(M1)+Pr(M2) - Pr(MinM2). O
A8.2 Let A be an m × n matrix and B be an n × p matrix. (a) Show that col(B) C null(A) if and only if AB = 0. (b) Show that if AB = 0, then rank(A) + rank(B) 〈 n. A8.2 Let A be an m × n matrix and B be an n × p matrix. (a) Show that col(B) C null(A) if and only if AB = 0. (b) Show that if AB = 0,...
0, P - 0 2 ) - 1. 3. Let (X, A,) be a complete measure space. Assume that A, B E A with (A) = (B) < . Show that if A CCCB, then CE A. 4 Let A and Rhe two collections of euheete of Y Aceume that any cot in 4 halanes
= 5a. (10 pts) Let fr : [0, 1] → R, fn(x) ce-nzº, for m = = 1, 2, 3, .... Check if the sequence (fn) is uniformly convergent. In the case (fr) is uniformly convergent find its limit. Justify your answer. Hint: First show that the pointwise limit of (fr) is f = 0, i.e., f (x) = 0, for all x € [0, 1]. Then show that 1 \Sn (r) – 5 (w) SS, (cm) - Vžne 1...
8. 2mn, b-m2-n, and c = m2 + n2 be the sides of a a. Let a Pythagorean triangle. Suppose that b -a + 1. Show that (m - n)2 - 2n2 1, and determine all such triangles. b. Find the smallest two such triangles. 8. 2mn, b-m2-n, and c = m2 + n2 be the sides of a a. Let a Pythagorean triangle. Suppose that b -a + 1. Show that (m - n)2 - 2n2 1, and determine...
Topology 3. Either prove or disprove each of the following statements: (a) If d and p map (X, d) X, then the identity topologically equivalent metrics (X, p) and its inverse are both continuous are two on (b) Any totally bounded metric space is compact. (c) The open interval (-r/2, n/2) is homeomorphic to R (d) If X and Y are homeomorphic metric spaces, then X is complete if and only if Y is complete (e) Let X and Y...
Please all thank you Exercise 25: Let f 0,R be defined by f(x)-1/n, m, with m,nENand n is the minimal n such that m/n a) Show that L(f, P)0 for all partitions P of [0, 1] b) Let mE N. Show that the cardinality of the set A bounded by m(m1)/2. e [0, 1]: f(x) > 1/m) is c) Given m E N construct a partition P such that U(f, Pm)2/m. d) Show that f is integrable and compute Jo...
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...