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In an example I am given that following function $F: \mathbb{R}^N \rightarrow \mathbb{R}^N$ is given by \begin{equation} F(x_1, x_2, ... , x_N)= \bigg( \sum\limits_{i=1}^{N} x_i^2 \bigg)(x_1,x_2,...,x_N) \end{equation} 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:
\begin{equation} 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} \end{equation}

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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.

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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.

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