# Notation - Functions in $\mathbb{R}^n$

In an example I am given that following function $$F: \mathbb{R}^N \rightarrow \mathbb{R}^N$$ is given by $$$$F(x_1, x_2, ... , x_N)= \bigg( \sum\limits_{i=1}^{N} x_i^2 \bigg)(x_1,x_2,...,x_N)$$$$ I am a bit confused with the notation. If I expand the function is it given by the following: $$F(x_1, x_2, ... , x_N)= x_1^2+x_2^2+...+x_N^2$$ ? I am given that it is a function from $$\mathbb{R}^N$$ to $$\mathbb{R}^N$$ so I think that is wrong but don't understand how the term $$(x_1,x_2,...,x_N)$$ at the end works.
Could it be that:
$$$$F(x_1, x_2, ... , x_N)= \begin{pmatrix} x_1^2+x_2^2+...+x_N^2 \\ x_1^2+x_2^2+...+x_N^2 \\ \vdots \\ x_1^2+x_2^2+...+x_N^2 \\ \end{pmatrix}$$$$

Let me write it down for you in the case $$N = 2$$. In that case, we have \begin{align} F(x, y) &= (x^2 + y^2)\cdot (x, y) & \text{The first paren is a number (scalar); the second is an ordered pair} \\ &= ((x^2 + y^2)x, (x^2 + y^2)y) & \text{Definition of scalar multiplication in \Bbb R^2} \\ &= (x^3 + xy^2, x^2y + y^3) & \text{A little algebra, not really necessary at all. } \\ \end{align}

Does that help?

You might ask "How do I know when a parenthesized expression is a number, or an ordered tuple, or simply the list of arguments to a function?" and the answer is, in part, "practice and experience."

In this case, you know that the codomain is $$\Bbb R^N$$, so you know that the stuff on the right hand side has to be an $$N$$-tuple. There's an $$N$$-tuple already visible (namely $$(x_1, \ldots, x_N)$$), but it's got this other thing in front...which you can then recognize as a scalar.

No. The $$j$$-th entry of the image is

$$x_j \sum_{i = 1}^N x_i^2$$

as the sum is just a scalar.