I am studying this article: http://arxiv.org/pdf/0906.4883.pdf
There is a little part that I do not understand, in the proof of theorem 5, page 4.
Let $P$ be the projection map of $L^p(\mathbb{R}^n)$ onto the linear span of the characteristic functions of the cubes $Q_i$
It says that the function $P$ defined before, is bounded and is actually s.t. $||P||=1$
Now how can I see this?
I think $P:L^p(\mathbb{R})\rightarrow T$ where $T\subset L^p(\mathbb{R})$ $T=\{\lambda_1\chi_{Q_1}+\lambda_2\chi_{Q_2}+..+\lambda_N\chi_{Q_N}: \lambda_i\in R\} $ and $\chi_i$ are the characteristic functions. But I don't really understand what norm they are considering for $P$, is it just the $L^p$ norm?
Thank you very much.