(c) Show that T-l is not bounded, and explain why this is not in contradiction with the Open Mapping Theorem and the Bounded Inverse Theorem.
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- 3 Show that the square integrable function f(x) = sin( πk log x/ log 2 )for k ≥ 1 are orthogonal over the interval 1 ≤ x ≤ 2 with respect to the weight function r(x) = 1/ x . Obtain the norms of the functions and construct the othornormal set.Consider the Cauchy Problem y 0 = a(x) arctan y, y(0) = 1, where a(x) is a continuous function defined on R, such that for every x it holds that |a(x)| ≤ 1. Using the Global Picard–Lindel¨of Theorem, show that there exists a unique solution y defined on R.Suppose that w and r are continuous functions on (−∞, ∞), W (x) is an invertible antiderivative of w(x), and R(x) is an antiderivative of r(x). Circle all of the statements that must be true.
- 1. Show (in terms of ε - δ) that a function f : R3 → R defined byf(x; y; z) = (2x + 3y + 4z) is uniformly continuous.Consider the Banach space C[0,1] of continuous functions on the interval [0,1] equipped with the sup-norm. Let T: C[0,1] -> C[0,1] be a bounded linear operator such that T(f) is continuously differentiable for every f in C[0,1]. Prove or disprove the following statement: "If T is injective, then T^{-1} is also bounded."Verify divergence theorem for f=x2i+zi+yzk taken over the cube bounded by x=0,x=1,y=0,y=1,z=0 and z=1
- Let f:X->Y be a function between metric spaces (X,d) and (Y,d). Prove that f: (0, infinity) -> R, f(x) is not uniformly continuous.Metrics. Show that, for fixed x0 in E, the function from E into ℝ+ is uniformly continuous.what is the global maximum and minimum of: f(x,y) = y^2 +xy -xy^2 in (0≤ x ≤ 1) and (1≤ y ≤ 2)
- 1.Suppose that f : [0, 1] −→ [0, 1] is a continuous function. Prove that f has afixed point in [0, 1], i.e., there is at least one real number x ∈ [0, 1] such thatf(x) = x. 2.The axes of two right circular cylinders of radius a intersect at a right angle.Find the volume of the solid of intersection of the cylinders.Show that f(x) = 1 x is not uniformly continuous on (0,1) but it is uniformly continuous on [1,2]The main point of this exercise is to use Green’s Theorem to deduce a specialcase of the change of variable formula. Let U, V ⊆ R2 be path connected open sets and letG : U → V be one-to-one and C2such that the derivate DG(u) is invertible for all u ∈ U.Let T ⊆ U be a regular region with piecewise smooth boundary, and let S = G(T). Solve all of them plz