Problem 29. Suppose 1 < p < 00 and f, g E IP(X, M,). Suppos lfllp...
3. Suppose that (M, ρ) is a compact metric space and f : (M, p)-+ (M,p) is a function such that (Vz, y E M) ρ (z, y) ρ (f (x), f (y)). a. Let x E (M, ρ) and consider the sequence of points {f(n) (X)}n 1 . (Remember: fn) denotes the composition of f with itself, n times, so for each n, f+() rn, k E N) such that ρ (f(m) (x) ,f(n +k) (r)) < ε ....
Suppose that (a-r, a) C E or (a, a + r) C E, f : E → R, L E R, and (1) Prove that there exist numbers 0 < δ < r and M > 0 such that If(x)| < M for all (2) Prove that if L is nonzero, then there exist numbers 0 < δ < r and η > 0 such that limx→af(x) = L xEEwith 0 < |x-a| < δ. If(x)| > η for all...
Please Complete 4.1. Exercises Exercise 4.1. Lete: G → GL(U), ψ: G → GL(V) and : representations of a group G. Suppose that Te HomG(φ, ψ) and Se Prove that ST Homc(p.,p). p: G GL(U Xp. Prove tha Exercise 4.2. Let o be a representation of a group G with character Exercise 4.3. Let p: GGL(V) be an irreducible representation Let be the center of G. Show that if a e Z(G), then p(a) Exercise 4.4. Let G be a...
where Problem 36. Assume f : X → [0, oo]. Prove that if Σ f(x) < 00, then {x E X (z) > 0} is a countable set. (HINT: Show that for every k E N the set {x E X | f(x) > k-1} is finite.) f(x)-sup f(x) | F is any finite subset of X TEF Problem 36. Assume f : X → [0, oo]. Prove that if Σ f(x) 0} is a countable set. (HINT: Show that...
5. (a) (5 points) Let R F[x] for a field F. Let f, g E R be nonzero. Prove that (f(x)) = (g(x)) if and only if g(x) = af(x) for some constant a E F. (b) (5 points) Let R be any ring. Prove that the nilradical Vo is contained in the intersection of all prime ideals.
3. (a) Suppose f : (a, b) + R is differentiable, and there exists M E R such that If'(x) < M for all x € (a, b). Prove that f is uniformly continuous on (a, b). (b) Let f : [0, 1] → [0, 1] be a continuous function. Prove that there exists a point pe [0, 1] with f(p) = p.
(8) Let E c R" and G C Rm be open. Suppose that f E -G and g:GR', so that h -gof:E R'. Prove that if f is differentiable at a point x E E and if g is differentiable at f(x) є G, then the partial derivatives Dh,(x) exist, for all , SO , . . . , n, and and J-: に1 The subscripts hi, 9i, k denote the coordinates of the functions h, g, f relative to...
(8) Let E C R" and G C R" be open. Suppose that f E G and g G R', so that h = go f : E → R. Prove that if f is differentiable at a point x E E, and if g is differentiable at f (x) E G, then the partial derivatives Dihj(x) exist, for all and j - ...., and 7m に! (The subscripts hi. g. fk denote the coordinates of the functions h, g....
(12) Suppose that f: [0, o0) - (0, 00) and that f e R((0, n]), for every n E N. Prove that f is Lebesgue measurable, the Lebesgue integral Jo.0)f dA exists, and f dA [0,00) lim f (x)dx noo (12) Suppose that f: [0, o0) - (0, 00) and that f e R((0, n]), for every n E N. Prove that f is Lebesgue measurable, the Lebesgue integral Jo.0)f dA exists, and f dA [0,00) lim f (x)dx noo
10. Suppose that f(x) E Z[x] and f(x) is irreducible over Z, where p is a prime. If deg f(x) n, prove that 2,[x]/f(x)) is a field with p" elements. 11. Construct a field of order 25.