Ricardo A. Sáenz

Due date: November 24 Problem 1 If the measurable $f_n\searrow f\ge 0$ with $\int f_1 < \infty$, then $\int f_n \to \int f.$ Explain the condition $\int f_1 < \infty$. Problem 2 Let $f\in L^1(\R)$ and $f^\delta(x) = f(\delta x)$. Then $\displaystyle \int f^\delta = \frac{1}{\delta}\int f.$ Problem 3 There exists a positive continuous $f \in L^1(\R)$ such that $\limsup_{|x|\to\infty} f(x) = \infty.$ If $f\in L^1(\R)$ is uniformly continuous, then $\lim_{|x|\to\infty}f(x) = 0.$ Problem 4 If $f\in L^1(\R)$ and $F(x) = \int_{-\infty}^x f$. Then  F is uniformly continuous.

Due date: November 17 Problem 1 For each $n\in\Z$, let $e_n(x) = e^{2\pi i nx}$. Then $\displaystyle \int_0^1 e_n(x) \overline{e_m(x)} dx = \begin{cases} 1 & n=m\\ 0 & n\not=m. \end{cases}$ Problem 2 For $f\in C([0,1])$, the sequence $\widehat f(n) \to 0$ as $|n|\to\infty$. Problem 3 Let $f\in C^1([0,1])$ with $f(0)=f(1)$. $\widehat{f'}(n) = 2\pi i n \widehat f(n)$ The Fourier series of  f converges uniformly to  f . Problem 4 Let $E\subset \R$ and $U_n$ the open set $U_n = \{x\in\R: d(x,E)<1/n\}.$ If  E is compact, $|E| = \lim |U_n|$. However, the previous conclusion may be false if either  E is closed and unbounded, or bounded and open. Problem 5 Let  E be the subset of $[0,1]$ of numbers which do not have the digit 4 in their decimal expansion. Find $|E|$.

Due date: November 10 Problem 1 Let  Y be a vector subspace of the normed space  X . Then its closure $\bar Y$ is also a vector subspace of  X . Problem 2 Let $X=C([-1,1])$ with the inner product $\displaystyle (f,g) = \int_{-1}^1 f\bar g.$ Apply the Gram-Schmidt process to the sequence $f_n(x) = x^n$ to obtain the orthonormal polynomials $p_0, p_1, p_2, p_3$, such that each $p_n$ is of degree  n . These are the first  Legendre polynomials . Problem 3 Let  Y be a close subspace of the Hilbert space  X and $T:X\to Y$ the orthogonal projection onto  Y , $Tx = \text{Proj}_Y x.$ T is continuous. Problem 4 Let  Y be a closed subspace of the Hilbert space  X , and let $Y^\perp = \{ x\in X: x\perp Y\}.$ $Y^\perp$ is a closed subspace of  X . $X\cong Y\oplus Y^\perp$. Problem 5 Let X be an inner product space and $\bar X$ its completion. $\bar X$ is a Hilbert space. If  X is separable, so is $latex \ba...