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I have some questions on functions that take real-valued vectors as arguments which I would like to resolve.

Given $x \in \mathbb{R}^a, y\in \mathbb{R}$. I know the notation $g: \mathbb{R}^a \rightarrow \mathbb{R}$, which means that $g$ is a function that takes a vector of size $a$ as an argument and outputs a scalar.

But what if I have something like $z = g(x,y)$ with $x \in \mathbb{R}^a, y\in \mathbb{R}^b, z\in \mathbb{R}$? What would be the correct notation?

Something like $g:\mathbb{R}^a \times \mathbb{R}^b \rightarrow \mathbb{R}$? Or $g:\mathbb{R}^{a+b} \rightarrow \mathbb{R}$? Or something different?

And is that the same btw? Because by the rules of exponentiation, it should be the same but how can one then distinguish between a function that takes two vectors, one of size $a$ and one of size $b$ or a function that just takes one vector, but that one of size $a + b$?

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