I need to show that if matrix A satisfies the RIP then it also satisfies the RNP. I need to prove that each of the lemmas holds and then show that RIP implies RNP using the lemmas.
Lemma 1: Let the vector have nonzero entries. Then $$ \|x\|_1 \leq \sqrt{s} \|x\|_2 $$ where x is an s-sparse vector.
Lemma 2: Let $u,v$ be $s$-sparse vectors. If $\operatorname{supp}(u) \cap \operatorname{supp}(v) = \emptyset$, then $$ \frac 1m |\langle Au, Av| \leq \delta_{2s}\|u\|_2\|v\|_2 $$ where $\delta_{2s}$ denotes the constant assocaited with the RIP of order $2s$.
What I've managed: $\frac{1}{m}|⟨, ⟩| = \frac{1}{m}|v^T A^T A u|$ $= |v^T \frac{1}{m}A^T A u| = |v^T (\frac{1}{m}A^T A - I)u|$
Alternative method: $\frac{1}{m}|⟨, ⟩| = \frac{1}{m}|⟨, ⟩ - ⟨, ⟩| $ But I don't fully understand how that part is explained. Could someone show how it's done?
Lemma 3: Let $a,b$ be two $s$-dimensional vectors such that the largest entry in the absolute value of $b$ is no larger than the smallest entry in the absolute value of $a$. Then $$ \|a\|_1 \geq \sqrt{s} \|b\|_2 $$
I especially need help proving the lemmas.