Let's consider two vectors $\mathbf{u}$ and $\mathbf{u}$, which are l1 normalized, where $u, v \in \mathbb{R}_+^n$. I would like to know if there is an upper bound on the dot product of these two vectors.
I think the dot product of two vectors that are l2 normalized will be less than one, as in this case the dot-product is the length of the projection of one vector onto the other. I am not sure if I can use the same reasoning when considering the l1 norm. In case it is, can we somehow prove that, $\mathbf{u} \cdot \mathbf{v} \le 1$ where $\|\mathbf{u}\|_1 = \|\mathbf{v}\|_1 = 1$.