Let $E_1 ,..., E_n $ be a sequence of disjoint Borel sets in $\mathbb{R} ^n $ of positive finite measure and let $\chi_1,...,\chi_n$ be their characteristic functions.
Given $1 \leq p < \infty$ , prove that the operator $P$ on $ L ^p (\mathbb{R}^n ) $ defined by: $Pf: = \sum_{r=1}^n |E_r|^{-1} \langle f, \chi_r \rangle \chi_r $ is a projection of finite rank, and find its norm and range.
I was wondering what I need to prove here... What should I prove in order to say this is a projection? Afterwards, how can I prove that the dimension of the range of such an operator is finite?
Thanks in advance