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 $\|T \|$?
4. Read about Hölder inequality and Minkowski inequality. In the simplest setting, we have
$$ (\sum_{i=1}^n |x_i y_i|) \leq (\sum_i |x_i|^p)^{1/p} (\sum_i |y_i|^q)^{1/q} $$
Read about the proof (in wiki, or any textbook about functional analysis, say Folland). Why it works?