an inequality on $L_p$ and $l_2$

Question

Let $\{{f_i}\}$ be a countable or finite collection of good functions (e.g. Schwartz functions on $\mathbb{R}$). Let $1<p\le2$. Is it true that $$\|(\sum_i|f_i|^2)^{\frac{1}{2}}\|_{L^p}\le(\sum_i\|f_i\|_{L^p}^p)^{\frac{1}{p}} $$

Answer 1

The inequality follows from the fact that if $(a_i)$ is a countable family of non-negative numbers, then $$\left(\sum_{i\in\mathbb N}a_i^2\right)^{1/2}\leqslant \left(\sum_{i\in\mathbb N}a_i^p\right)^{1/p}.$$