Due date: December 7 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 For $latex r\in\mathbb Z_+$, let $latex \mathscr P_r$ be the space of polynomials of degree at most $latex r$. If $latex f_n\in\mathscr P_r$ converge uniformly to f in [0,1], then $latex f\in\mathscr P_r$. 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 I