Here is a quote from pages 4 and 5 of https://aetos.it.teithe.gr/~gouliana/en/2017-polynomials.pdf
As it is well known from nonlinear algebra, the structure of a typical nonlinear algebraic system of $n$ equations with $n$ unknowns has the form $$f_1(x_1, x_2, ..., x_n)=0$$ $$f_2(x_1, x_2, ..., x_n)=0$$ $$\cdots$$ $$f_n(x_1, x_2, ..., x_n)=0$$ Or using vector notation, $F(x)=0$ where $$F=(f_1, f_2, ..., f_n)^T$$ is a vector of non-linear functions $f_i(x)=f_i(x_1, x_2, ..., x_n)$ each being defined in the vector space $$\Omega=\prod_{i=1}^{n}{\{\alpha_i, \beta_i\} \subset \text{R}^n}$$ of all real valued and continuous functions.
My question is what does the above vector space mean?
From the context, he means that each of the coordinate functions $f_i$ is defined and continuous on $$\Omega=\prod_{i=1}^{n}{[\alpha_i, \beta_i] \subset \text{R}^n}$$
The vector space referred to is the vector of space of such functions.
To put it another way, $$ \Omega=\{(x_1,x_2,\dots,x_n)|\alpha_i\leq x_i\leq\beta_i, i=1,2,\dots,n\} $$
This isn't what the paper says, but I feel pretty sure that's what it means. I'm not certain, though, about the inequalities. Perhaps they are meant to be strict.