Homework 13, Real Analysis

Due date: November 3

Problem 1

Let f_n(x) = a_n x^2 + b_n x + c_n be a sequence of quadratic polynomials such that

\displaystyle \int_0^1 |f_n(x)| dx \to 0.

Then the coefficient sequences a_n, b_n, c_n all converge to zero.

Problem 2

  1. For r\in\Z_+, let \mathscr P_r be the space of polynomials of degree at most r. If f_n\in\mathscr P_r converge uniformly to f in [0,1], then f\in\mathscr P_r.
  2. The polynomials \displaystyle f_n(x) = 1 + \frac{1}{2}x + \frac{1}{2^2}x^2 + \ldots + \frac{1}{2^n} x^n converge uniformly on [0,1], but their limit is not a polynomial function.

Problem 3

Let \mathscr H be the subspace of functions f\in C([0,1]) that satisfy

f(1 - x) = f(x).

Then \mathscr H is an infinite dimensional closed subspace of C([0,1]).

Problem 4

Let \mathscr I: C([0,1])\to C([0,1]) be the operator given by

\displaystyle \mathscr If(x) = \int_0^x f(t) dt.

  1. \mathscr I is continuous with respect to the uniform norm.
  2. If f_n,f\in C([0,1]), f_n\rightrightarrows f, F_n(x) = \int_0^x f_n and F(x)=\int_0^x f, then F_n\rightrightarrows F.

Problem 5

Use the previous problem to prove the following theorem: Let f_n\in C^1([0,1]) such that

  • f_n(x_0) converges for some x_0\in[0,1];
  • f_n'\rightrightarrows g.

Then f_n converges uniformly and, if f_n\rightrightarrows f, then f \in C^1([0,1]) and f'=g

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