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Functional Analysis

Space with a Universal Covering Space

Let X be a space which has a universal covering space. If (X1, p1) is a covering space of X and (X2, p2) is a covering space of X1, then (X2, p1p2) is a covering space of X. See the attached file.

Functional Analysis

Attachment file. Let X be a normed space and . Show that if for every bounded linear functional f on X , then .

Graphically Analysis Trigonometric Functions

Please explain step by step how to complete the following: In the following determine graphically whether the equation could possibly be an identity. If it could, prove that it is. 5. sin^4 t - cos^4 t = 2 sin² t -1 ans. - sin^4t - cos^4 t = (sin²2t - cos²t) (sin²t + cos²t) = (sin²t-(1-sin²t)) (1) = 2 sin²t -

Volume of a solid of revolution about the Y axis

Find the volume of the solid generated by revolving the region described about the Y axis: x=e^y intersecting the x axis at (1,0) and between the points on the y axis (0,0) and (0,ln3) Using the formula V=∫π[R(y)]²dy

Graph Parabolas Points

24. Graph the following parabola clearly labeling exact points for the vertx, x-intercepts(s), and y-intercept(s): y=2xsquare+4x-3

Uniform Convergence

Please see the attached file for the fully formatted problems. We are using the book Methods of Real Analysis by Richard R. Goldberg.

A Discussion On Rolle's Theorem

Please solve the following with as much explanation of each step as possible. Let f be continuous on the closed interval [a,b] and differentiable on the open interval (a,b). Also, suppose that f(a) = f(b) and that c is a real number in the interval such that f'(c) = 0. Find an interval for the function g over which Rolle's

Find number of degrees...

Use the equation s=rO for the following: 1. Find the number of degrees in the central angle of a circle with radius of 20 inches if the angle subtends an arc of 13 inches. 2. Find the length of a degree on the equator considering the diameter of the equator to be 7912 miles. 3. A bicycle has a 28 inch wheel. How man

Outer Measure Functions

Let A = union ( i from 1 to infinity) of M_i, Mi's are disjoint, show that m*(A) = sum (i from 1 to infinity) of |M_i| m*(A) is the outer measure of A, that is, m*(A) = inf sum (i from 1 to infinity) of M_i. PLEASE NOTICE THE = SIGN, A = the union, not a subset of the union.

Intermediate value theorem

Determine if the following conforms to the intermediate value theorem. Clearly give the conditions and, if appropriate, find an appropriate value for c. f(x)=10/((x^2)+1) [0,1] k=8

Weak Convergence

Please explain why the following sequence for otherwise is an example of a sequence in such that weakly, but not strongly. Please see the attached file for the fully formatted problem.

Volumes of revolution

Find the volume of the object rotated by the given points. 1. y=1/x, y = 0, x = 1, x = 3; about y = -1 2. y = x, y = 0, x = 2, x = 4; about x = 1

Poles & Singularities

Classify the behavior at infinity (analytic, pole, zero, or essential singularity; if a zero or pole, give its order) of the following functions: f1(z) = (z^3 + i)/z f2(z) = e^(tan(1/z))

Find a horizontal line that divides an area into two equal parts.

Find a horizontal line that divides an area between y=x^2 and y=9 into two equal parts. I need help in approaching this problem. I know how to calculate the area. I divided the area by two and through trial and error I found the horizontal line to be close to y=5.6. I don't know how to get an exact answer.

Polar coordinates

X and y represents rectangular coordinates. What is the given equation using polar coordinates (r, theta). x^2 = 4y

CalcII

The resale value os a certain idustrial machine decreases over a 10 year period at a rate that changes with time, when the machine is x years old, its value is decreasing at a rate of 220(x-10) dollars/year. By how much does the machine depreciate during the second year? Please see attached for full question.

Area, volume, and expantion

A.) let A be the area of a circle of radius r that is changing w/ respect to time. if dr/dt is a constant, is dA/dt a constant, explain. b.) let V be the volume of a sphere of radius r that is changing w/ respect to time. If dr/dt is constant, is dV/dt constant, explain. c.) All edges of a cube are expanding at a rate of