Real Analysis : Bounded Continuity / Differentiability

Problem: Let f: [0, ∞) → R be a bounded function. For all X greater than or equal to 0, let G(x)=sup{f(t): 0 is less than or equal to t is less than or equal to x} a) Show that if f is continuous, g is also continuous. Is the converse also true? Justify. b) If f is differentiable and continuous, is g also d ...continues

Real Single-Variable Analysis Problem

Let x be an irrational number. Prove that there exist infinitely many fractions (p/q) with p and q as integers such that: abs(x-[p/q]) < 1/(q^2)

Real Analysis : Using a Summation Series to Estimate an Integral

Say the only tool you have available to you is a pocket calculator which performs addition, subtraction, multiplication, and division, accurate to 15 decimal places. Explain a practical way to compute: Integral from 0 to 1 of e^[-(x^2)] to within an error less than 10^-8. Prove that the method works.

Real Analysis : Subintervals

Prove rigorously: Let N be an integer > or equal to 2, and let Xsub0....Xsubn E [0,1). Prove that there exist i and j with i not equal to j such that abs (xsubi-xsubj) < 1/n.

Real Analysis -- Real Numbers/Integers Theory

Let x be a real number, and let N be an integer ≥ 2. Prove that there exist integers P and Q such that: 1 ≤ q ≤ N and absolute value of [x-(P/Q)] < 1/(QN)

Using a Summation Series to Estimate a Quantity

Say the only tool given to you is a calculator which performs addition, subtraction, multiplication, and division. Let X= Summation (k=1 -->n) e^-(k/n)^2 with N^20 Explain a practical way of computing X within an error of 10^8. Roughly how big is X?

Inner Product Space Convergence

Problem. If is a bounded sequence in an inner product space, and is a sequence converging to zero, prove that . Note, here < , > is the inner product notation. Hint: Use triangle inequality.

Show that 2 sequences are orthonormal

Problem. Show that the following sequences are orthonormal. (a) in the space . (b) in the space .

Hilbert Space and Subspace

Problem. Show that if is an orthonormal set in a Hilbert space H, then the set of all vectors of the form is a subspace of H. Hint: Take a Cauchy sequence , where . Set and show that is a Cauchy sequence in . Please see the attached file for full problem description.

Functional Analysis - Normed Linear Space

Suppose that N is a normed linear space. Prove that each finite dimensional linear submanifold of N is complete and therefore closed.