$1\leq p < \infty$. Space is $L^p(\mathbb{R}^n)$.
Let $\delta >0,\ R>0$ be constants. $Q$ is the open cube centered at origin such that $||y||<\frac{\delta}{2}, \forall y \in Q$.
Let $Q_1, \dots, Q_N$ be mutually non overlapping translates of $Q$ such that $B(0,R)\subset \bigcup_{i=1}^N Q_i $.
Define projection map, $P:L^p(\mathbb{R}^n) \rightarrow L^p(\mathbb{R}^n)\ $ as
$$Pf= \sum_{i=1}^N \bigg( \frac{1}{|Q_i|} \int_{Q_i}f(z)dz \bigg) \chi_{Q_i}$$
where, $|Q_i|=$ Lebesgue measure of $Q_i$ ; $\chi_{Q_i}$ is the characteristic function of $Q_i$.
I have to show that $||P||=1$.
Can this be done by the definition that, $||P||= Sup_{||f||_p \leq 1} ||Pf||_p$? I am getting stuck because there is a double integration involved. Any help is appreciated.
Look at what each term in the sum is: $\left(\frac{1}{|\mathcal{Q}_i|}\int_{\mathcal{Q}_i}fdx\right)\chi_{\mathcal{Q}_i}$ is a step function taking the average value of $f$ over $\mathcal{Q}_i$ in $\mathcal{Q}_i$ and zero everywhere else--in particular, zero on all other $\mathcal{Q}_j$s. Perhaps for ease of notation you should write $c_i=\int_{\mathcal{Q}_i}fdx$. The "double integral" as you say will only pick out a factor of $|\mathcal{Q}_i|$ in each term in the sum.
Think about (i'll leave the details to you, they're not hard) how the facts I mentioned yield the following equality: $$\|Pf\|^p_{L^p(\mathbb{R}^n)}=\sum_{i=1}^N\frac{|c_i|^p}{|\mathcal{Q}_i|^{p-1}}.$$
Now here's where it is less obvious where to proceed, but motivated by the fact that if $f$ is non-negative, the $|c_i|$'s are just the $L^1$ norms of $f$ restricted to the respective boxes, and using this nice embedding, we get
$$\|Pf\|^p_{L^p(\mathbb{R}^n)}\leq\sum_{i=1}^N\|f|_{\mathcal{Q}_i}\|^p_{L^p(\mathcal{Q}_i)}\leq\|f\|^p_{L^P(\mathbb{R}^n)}.$$ From here, think about how we might achieve equality?