Assume $\boldsymbol{w}\in \mathbb{R}^{M}$ to be a vector. How, can I express
$$l_1=\sum_{m=1}^{M}|w_m|$$ using only vector notation?
I essentially want to express the $\ell^1$ norm without introducing a new notation to my readers, who are not mathematicians.
I thought about introducing the vector $\tilde{\boldsymbol{w}}$ as the vector which has the absolute values of the coordinates of $\boldsymbol{w}$ as its coordinates and then to use the vector $\boldsymbol{1}=[1,1,\ldots,1]^T\in\mathbb{R}^M$ to write
$$l_1 = \boldsymbol{1}^T\tilde{\boldsymbol{w}}$$
but this seems quite involved.
EDIT: I also thought about introducing the vector valued component wise signum function $\textbf{sgn} (\boldsymbol{w})$ and then to use
$$l_1 = \textbf{sgn}^T (\boldsymbol{w})\boldsymbol{w}.$$
Is there a better way to do this?
Edit: I don't want to use $\|\boldsymbol{w}\|_1$ because I will have to define this by using the sum itself.