Consider a function $f(x,y): A \subset \mathbb{R}^2 \to \mathbb{R}$ and the series of function
$$\sum_{n \geq 0} f_n(x,y)\tag{1}$$
How is unform convergence defined for $(1)$?
Is it defined a uniform convergence "with respect to x", as y was just a parameter, as follows?
$$\lim_{N \to \infty} \bigg[\sup_{x \in A}\Bigg|\sum_{n \geq 0} f_n(x,y) - \sum_{n = 0}^{N} f_n(x,y)\Bigg| \bigg]=0$$ And analogous for $y$?
If it is, then are the theorems of switching series and integrals (not double integrals but just inegrals in $dx$ or $dy$), series and derivatives (with respect to $x$ or $y$) and series and limits (on $x$ or $y$) still valid if the convergence of $(1)$ is uniform with respect to the interseted variable?