# 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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