Homework 13, Real Analysis

Due date: November 3

Problem 1

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

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

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

Problem 2

1. For $latex r\in\Z_+$, let $latex \mathscr P_r$ be the space of polynomials of degree at most r. If $latex f_n\in\mathscr P_r$ converge uniformly to f in [0,1], then $latex f\in\mathscr P_r$.


2. The polynomials $latex \displaystyle f_n(x) = 1 + \frac{1}{2}x + \frac{1}{2^2}x^2 + \ldots + \frac{1}{2^n} x^n$ converge uniformly on [0,1], but their limit is not a polynomial function.

Problem 3

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

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

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

Problem 4

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

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

1. $latex \mathscr I$ is continuous with respect to the uniform norm.


2. If $latex f_n,f\in C([0,1])$, $latex f_n\rightrightarrows f$, $latex F_n(x) = \int_0^x f_n$ and $latex F(x)=\int_0^x f$, then $latex F_n\rightrightarrows F$.

Problem 5

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

• $latex f_n(x_0)$ converges for some $latex x_0\in[0,1]$;


• $latex f_n'\rightrightarrows g$.

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