Let $n\ge 1$ and $a=(a_1,\dots, a_n)\in\mathbb{R}^n$ be a nonnull vector. There is an upper bound for the sum of the magnitude of each component of the vector $a$? I mean, for $$\sum_{i=1}^n |a_i|?$$
I would say that $$\sum_{i=1}^n |a_i|\le |a|$$
but I am not sure about that. Here $|a|$ denotes the magnitude of the vector $a$, which is given by $$|a|=\sqrt{\sum_{i=1}^n |a_i|^2}.$$ Also, I am interested other bounds (maybe finer, if any).
Thank you in advance.
This is wrong. Let $a = (3, 4)^\top$: $$ \sum_{k=1}^2 \lvert a_k \rvert = 7 > \sqrt{\sum_{k=1}^2 \lvert a_k\rvert^2} = 5 $$ What you can do for any $x \in \mathbb{R}^n$. Let $\lvert x_i \rvert = \max_{k \in \lbrace 1, ..., n \rbrace} \lvert x_k \rvert$ is the following: $$ \sum_{k=1}^n \lvert x_k \rvert \leq n \lvert x_i \rvert = n \sqrt{\lvert x_i \rvert^2} \leq n \sqrt{ \sum_{k=1}^n \lvert x_k \rvert^2 } = n \lvert x \rvert $$