HW 8

1. Read appendix F about Littlewood's three principles, and write some comments about it in your webpage (for example, a summary of what this is about, or questions)

2. Do Pugh Ex 83

3. Let $(\R^n, | \cdot |_{1})$ be the normed vector space where $|(x_1, \cdots, x_n)|_{1}: = \sum_i |x_i| $. Let $T: \R^n \to \R^n$ be a linear operator, given by the matrix $T_{ij}$, that sends $(x_i)$ to $(y_j)$, where $y_i = \sum_j T_{ij} x_j$. How to compute $\|T \|$?

• optional: if we use $\| - \|_{max}$ norm on $\R^n$, how to compute the operator norm $\|T\|$?

4. Read about Hölder inequality and Minkowski inequality. In the simplest setting, we have

• (Hölder inequality), for $p,q \geq 1$ that $1/q+1/p=1$, we have

$$ (\sum_{i=1}^n |x_i y_i|) \leq (\sum_i |x_i|^p)^{1/p} (\sum_i |y_i|^q)^{1/q} $$

• (Minkowski inequality) for any $p\geq 1$, $$(\sum_{i=1}^n |x_i + y_i|^p)^{1/p} \leq (\sum_i |x_i|^p)^{1/p} + (\sum_i |y_i|^p)^{1/p} $$

Read about the proof (in wiki, or any textbook about functional analysis, say Folland). Why it works?