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- Let (X, T) and (Y, T1) be two topological spaces and let f be a continuous mapping of X into Y. If (Y, T1) is a T1 space, then (X, T) is a T1 space?Let (X1, d1) and (X2, d2) be separable metric spaces. Prove that product X1 × X2 with metric d((x1, x2), (y1, y2)) = max{d1(x1, y1), d2(x2, y2)} is also separable space.1. a) Let (x, d) be a metric space. Define a flow on (x, d). b) Let (x, {ϕt}) be a flow on a metric space X. When is xo in x a fixed point of the flow? c) When do you say that a fixed point xo in x is Poincare stable? d) When do you say that a fixed point xo is Lypanov stable?
- Let f(x) = 21 - x2 and g(x) = x2 + 3. Use symmetry, if appropriate, to help find the center of gravity, ( x, y ), of the bounded region enclosed by the graphs of f and g.Let V be an inner product space, and let y, z ∈V. Define T: V →V by T(x) = <x, y>z for all x ∈V. First prove that T is linear. Then show that T∗exists, and find an explicit expression for it.Let (R>0, d) be the metric space defined by d(x, y) =|log (y/x)|. This metric space is isometric to the Euclidean line E1, where an isometry E1 → (R>0, d) is given by x→ ex . proof that x→ ex is isometric.
- 2. Assume that (X, dX) and (Y, dY ) are complete spaces, and give X × Y themetric d defined byd((x1, y1),(x2, y2)) = dX(x1, x2) + dY (y1, y2)Show that (X × Y, d) is complete.Find the absolute maximum and minimum of f(x,y)= 4xy^2 - (x^2)(y^2) - xy^3 on the closedtriangular region with vertices (0,0), (0,6), and (6,0).Prove that topological space E is not homeomorphic to the spaceY = {(x, y) ∈ E^2 : y = ± x} (E represents R equipped with Euclidean distance, E^2 represents R^2 equipped with euclidean distance)
- 2-What is the size of the vector space consisting of polynomials of degree not exceeding n? A) 0 B) 2n+1 C) n-1 D) n+1 E) nCompute the flux os F across S. F = < 2xz, 5y2, -z2 > The enclosed surface S is bounded by z = y , z = 4, z = 2 - 1/2x2, x = 0, z = 0.Let f: X Y be a continuous surjection between metric spaces. If X is compact then ............................. A. X is complete B. Y is connected. C. Y is not compact. D. Y is not neccessarily complete.