Ricardo A. Sáenz

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 \disp...

Due November 30 Problem 1 Let  Y be a finite dimensional subspace of the Hilbert space  X . Then  Y is closed in  X . Problem 2 For each $latex n\in\Z$, let $latex e_n(x) = e^{2\pi i nx}$. Then $latex \displaystyle \int_0^1 e_n(x) \overline{e_m(x)} dx = \begin{cases} 1 & n=m\\ 0 & n\not=m. \end{cases}$   Problem 3 Let  Y be a close subspace of the Hilbert space  X and $latex T:X\to Y$ the orthogonal projection onto  Y , $latex Tx = \text{Proj}_Y x.$ T is continuous. Problem 4 Let  Y be a closed subspace of the Hilbert space  X , and let $latex Y^\perp = \{ x\in X: x\perp Y\}.$ $latex Y^\perp$ is a closed subspace of  X . $latex X\cong Y\oplus Y^\perp$.

Problem 1 Let $latex \Omega\subset\R^n$ be a bounded domain and $latex u_0\in C(\bar\Omega)$. Suppose that $latex u\in C^{2,1}(\Omega\times(0,\infty))\cap C(\bar\Omega\times[0,\infty))$ is a solution of $latex \begin{cases} \partial_t u - \Delta u=0 & \text{in }\Omega\times(0,\infty)\\ u(\cdot,0) = u_0 & \text{on } \Omega \\ u=0 & \text{on } \partial\Omega\times(0,\infty).\end{cases}$ Then there exist constants $latex \mu, C>0$, depending only on $latex n,\Omega$, such that $latex \displaystyle \sup_\Omega |u(\cdot,t)| \le C e^{-\mu t} \sup_\Omega |u_0|$ for any $latex t>0$. Problem 2 Let $latex \Omega\subset\R^n$ be a bounded domain, $latex c\in C(\bar\Omega\times[0,T])$ with $latex c\ge -c_0$ for a constant $latex c_0\ge 0$, and $latex u_0\in C(\Omega)$ nonnegative. Suppose $latex u\in C^{2,1}(\Omega\times(0,T])\cap C(\bar\Omega\times[0,T])$ is a solution of $latex \begin{cases} \partial_t u - \Delta u + cu = -u^2 & \text{in }\Omega\times(0,T]\\ u(\cdot,0) = u_0...

Due November 23 Problem 1 Let  X be a metric space, $latex \mathcal C_X$ the set of its nonempty compact subsets and $latex d_H$ the Hausdorff metric. If  X is discrete, then $latex (\mathcal C_X,d_H)$ is discrete. Let $latex A\subset X$ be a finite set of isolated points of  X . Then  A is an isolated point in $latex \mathcal C_X$. Problem 2 Let $latex X=C([-1,1])$ with the inner product $latex \displaystyle (f,g) = \int_{-1}^1 f\bar g.$ Apply the Gram-Schmidt process to the sequence $latex f_n(x) = x^n$ to obtain the orthonormal polynomials $latex p_0, p_1, p_2, p_3$, such that each $latex p_n$ is of degree  n . These are the first  Legendre polynomials . Problem 3 If $latex ||\cdot||_1, ||\cdot||_2$ are norms induced by inner products in $latex \R^l$, then they are equivalent: there exist constants $latex c,C>0$ such that $latex c||x||_1 \le ||x||_2 \le C||x||_1$ for all $latex x\in\R^l$. ( Hint: use the Gram-Schmidt process to construct orthonrmal bases for each inner ...