6. In this problem, u is Lebesgue measure on R, while v is counting measure on...
(6) Let fel ), where is Lebesgue measure on R. Define F:R → R by F(x) = f' f(t) dx. (a) Prove that F is a continuous function. (b) Prove that F is uniformly continuous on R. (Note that R is not compact.)
Problem 6. Let V be a vector space (a) Let (--) : V x V --> R be an inner product. Prove that (-, -) is a bilinear form on V. (b) Let B = (1, ... ,T,) be a basis of V. Prove that there exists a unique inner product on V making Borthonormal. (c) Let (V) be the set of all inner products on V. By part (a), J(V) C B(V). Is J(V) a vector subspace of B(V)?...
(1 point) 5x2 — 5у, v %3D 4х + Зу, f(u, U) sin u cos v,u = Let z = = and put g(x, y) = (u(x, y), v(x, y). The derivative matrix D(f ° g)(x, y) (Leaving your answer in terms of u, v, x, y ) (1 point) Evaluate d r(g(t)) using the Chain Rule: r() %3D (ё. e*, -9), g(0) 3t 6 = rg() = dt g(u, v, w) and u(r, s), v(r, s), w(r, s). How...
Bartle The Elements of Integration and Lebesgue Measure:
4.R. If fe M*(X, X) and is due < +00, then the set N = {xe X: f(x) > 0} is o-finite (that is, there exists a sequence (Fa) in X such that N CU Fn and u(F.) < too).
equivalent 4. Let E C R. Prove that the following statements are (a) E is Lebesgue measurable (b) Given e> 0, there exist m* denotes the Lebesgue measure of a set (c) Given e 0, there exist a closed set F such that F C E and m* (E- F) < E. (d) There exists a set G (a countable intersection of open sets) such that E C G and m* (G - E) 0 (e) There exists a set...
equivalent 4. Let E C R. Prove that the following statements are (a) E is Lebesgue measurable (b) Given e> 0, there exist m* denotes the Lebesgue measure of a set (c) Given e 0, there exist a closed set F such that F C E and m* (E- F) < E. (d) There exists a set G (a countable intersection of open sets) such that E C G and m* (G - E) 0 (e) There exists a set...
equivalent 4. Let E C R. Prove that the following statements are (a) E is Lebesgue measurable (b) Given e> 0, there exist m* denotes the Lebesgue measure of a set (c) Given e 0, there exist a closed set F such that F C E and m* (E- F) < E. (d) There exists a set G (a countable intersection of open sets) such that E C G and m* (G - E) 0 (e) There exists a set...
Assume that is the parametric surface r= x(u, v) i + y(u, v) j + z(u, v) k where (u, v) varies over a region R. Express the surface integral 116.3.2) as as a double integral with variables of integration u and v. a (x, y) a(u, v) du dy ru Хry dy du l|ru Xr, || f (x (u, v),y(u, v),z (u, v)) 1(xu, Wsx,y,z) Mos u.v.gou,» @ +()*+1 li ser(u, v),y(u, v),z (u, v) Date f (u, v,...
1. Let u be the counting measure on P(N). Find a decreasing sequence of sets Ak such that -(4x) + lim y( s). AR) + lim (AK).
Problem 5. Given a vector space V, a bilinear form on V is a function f : V x V -->R satisfying the following four conditions: f(u, wf(ū, ) + f(7,i) for every u, õ, wE V. f(u,ū+ i) = f(u, u) + f(ū, w) for every ā, v, w E V. f(ku, kf (ū, v) for every ū, uE V and for every k E R f(u, ku) = kf(u, u) for every u,uE V and for every k...