Ricardo A. Sáenz

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

Due date: October 27 Problem 1 Let $latex x_n\to x, y_n\to y$ be convergent sequences in the normed space $latex (X,||\cdot||)$ over $latex \K$. Then $latex x_n + y_n \to x + y$; $latex \lambda_n x_n \to \mu x$ for all sequences $latex \lambda_n\to\lambda$ in $latex \K$. Problem 2 Let $latex (X,||\cdot||)$ be a normed space over $latex \K$. $latex \lim ||x_n-x|| = 0$ implies $latex \lim||x_n||=||x||$. For $latex x,y\in X,\lambda\in\R$, find $latex \lim \big( ||(n+\lambda)x + y|| - ||nx+y||\big)$. Problem 3 Let  X be a Banach space, $latex x_n\in X$, $latex ||x_n||=1$ for all n , and $latex \lambda_n\in\K$. Discuss the validity of the statement:  $latex \sum \lambda_n x_n$ converges if and only if $latex \sum |\lambda_n|<\infty$ . Problem 4 Let $latex x_n$ be a sequence in a Banach space  X such that, for all $latex \e>0$, there exists a convergent sequence $latex y_n$ such that $latex ||x_n - y_n||<\e$ for all  n . $latex x_n$ converges. Give an example where the...

Due date: October 20 Problem 1 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 $latex A_n\subset X$ be nonempty compact sets such that $latex A_{n+1}\subset A_n$. Then $latex \displaystyle A_n \to \bigcap_{k\ge 1} A_k$ in $latex (\mathcal C_X, d_H)$. Problem 3 Two norms $latex ||\cdot||_1, ||\cdot||_2$ are  equivalent if there exist constants $latex c_1, c_2>0$ such that $latex c_1 ||x||_1 \le ||x||_2 \le c_2 ||x||_1$ for all $latex x\in X$. If $latex ||\cdot||_1, ||\cdot||_2$ are equivalent, for all $latex \e>0$ there exist $latex \delta_1, \delta_2>0$ such that $latex B_{\delta_1}^1(x) \subset B_\e^2(x)$ and $latex B_{\delta_2}^2(x) \subset B_\e^1(x)$ for all $latex x\in X$, where $latex B_r^i(x)$ is the ball with respect to the norm $latex ||\cdot||_i$. If there exist $latex \delta,\e>0$ such that $latex B_\delt...

Due date: October 13 Problem 1 The IVP $latex \displaystyle\begin{cases} x'(t) = \sqrt{x(t)}\\ x(0)=0\end{cases}$ has an infinite number of solutions. Problem 2 If $latex A=\begin{pmatrix} 1/12 & 5/8\\ 5/8 & 1/12\end{pmatrix}$, then, for any $latex x\in\R^2$, $latex |Ax| \le \dfrac{1}{2}|x|$. Problem 3 Let $latex F(x,t) = \dfrac{tx}{x^2+1}$. Then, for all $latex t\in\R$, $latex |F(x,t) - F(y,t)| \le |t||x-y|$. Problem 4 The function $latex f(x) = \sqrt x$ on $latex [0,\infty)$ is uniformly continuous but not Lipschitz. Problem 5 Consider the operator $latex \Phi:C([-1,1])\to C([-1,1])$ given by $latex \displaystyle \Phi(x)(t) = 1 + 2 \int_0^t s x(s) ds,$ for any $latex x(t)\in C([-1,1])$. Starting from the constant function $latex x_0(s)=1$, verify explicitly that the  n th iteration of $latex x_{n+1} = \Phi(x_n)$ is the  n th Taylor polynomial of $latex t\mapsto e^{t^2}$ around $latex t=0$.

Due date: October 6 Problem 1 If $latex A\subset \R^n$ is convex, then $latex \bar A$ is convex. Problem 2 State whether the following are true or false. If $latex A,B$ are path connected, then $latex A\cap B$ is path connected. If $latex A, B\subset\R^n$ are convex, then $latex A\cap B$ is convex. Problem 3 Let $latex A\cap B\not=\emptyset$ in some metric space. State whether the following are true or false. If $latex A,B$ are path connected, then $latex A\cup B$ is path connected. If $latex A,B\subset\R^n$ are convex, then $latex A\cup B$ is convex. Problem 4 The fixed points of a continuous $latex f:\mathbb B^n\to\mathbb B^n$ might not be interior. The Brouwer fixed point theorem is false for the open ball.