For $x, y \in \mathbb{R}^n$, and assuming $x \neq 0$, my textbook says that $y$ can be written as
$ y = \frac{x}{\|x\|} (\frac{x}{\|x\|}.y) + (y-\frac{x}{\|x\|} (\frac{x}{\|x\|}.y))$
where ${\|x\|}$ is the Euclidean norm of $x$.
As far as I understand, the notation $\frac{x}{\|x\|}.y$ represents the dot product of the vectors $\frac{x}{\|x\|}$ and $y$, which is some real number $d$. The expression $\frac{x}{\|x\|}.y$ is written inside brackets, which means that the vector $\frac{x}{\|x\|}$ is being "multiplied" by $d$, but I don't know any function from $\mathbb{R}^n \times \mathbb{R}$ to $\mathbb{R}^n$ that defines such a "multiplication".
So what am I misunderstanding here?
For $c\in\Bbb R$ and $v\in\Bbb R^n$, both $cv$ and $vc$ (we'd usually write the former) are defined as the $w\in\Bbb R^n$ satisfying $w_i=cv_i$ for $1\le i\le n$.