Dynamical limitation and minimization I come across the following question: Under what conditions (on the series of functions $f_n$ or perhaps the domain of minimization) the following holds
$$
\lim_{n\to\infty}\min_{x_1,\ldots,x_n}f_n(x_1,\ldots,x_n) = \min_{\left\{x_i\right\}_i}\lim_{n\to\infty}f_n(x_1,\ldots,x_n)
$$
I can show that the above holds true when the minimization is carried over a finite number of variables independently of $n$, namely,
$$
\lim_{n\to\infty}\min_{x_1,\ldots,x_M}f_n(x_1,\ldots,x_M) = \min_{x_1,\ldots,x_M}\lim_{n\to\infty}f_n(x_1,\ldots,x_M)$$
But, I'm not sure how to approach the first question.
Thanks
 A: Let's a functions $F_n:\mathbb{X}\subset\mathbb{R}^{\mathbb{N}}\to \mathbb{R}$ such that $F_n(x_1,\ldots, x_n,\ldots )=f_n(x_1,\ldots, x_n )$ for all $f_n:\Pi_n(\mathbb{X})\subset\mathbb{R}^n\to \mathbb{R}$ of sequence $\{f_n\}_{n\in\mathbb{N}}$.  Here $\Pi_n$ is the projection $(x_i)_{i\in\mathbb{N}}\mapsto (x_1,\ldots, x_n)$. Most of the times when the minimum of a function $F:\mathbb{X}\subset\mathbb{R}^{\mathbb{N}}\to\mathbb{R}$ is achieved can admit the following conditions.


*

*Condition: supose there is $x^*_n=(x_1^*,\ldots, x_n^*,\ldots)$ such that  $$F_n(x^*_n)=\min_{x\in \mathbb{X} }F_n(x).$$

*Condition: supose thare is a sequence $\mathbb{R}^{\mathbb{N}} \ni x_{nk}\to x_{n\infty}\in \mathbb{R}^{\mathbb{N}}$ such that $$\lim_{k\to \infty}F_n(x_{nk})=F_n(x_{n\infty})=\min_{x\in \mathbb{X} }F_n(x).$$
Under these conditions the problem can be stated as
$$
\lim_{k\to \infty}\lim_{n\to \infty}F_n(x_{nk}) \mathop{=}^{?}\lim_{n\to \infty}\lim_{k\to \infty}F_n(x_{nk})
$$
For exemple for $\mathbb{X}\subset\ell^p$ whit usal topology of norm $\|\,\cdot\,\|_p$, we could to use the classical theorem 

Theorem. Let $\{ F_t ; t\in T\}$ be a family of functions $F_t : Y \rightarrow \mathbb{C}$ depending on a parameter t; let $\mathcal{B}_X$ be a base $Y$ and $\mathcal{B}_{T}$ a base in $T$. If the family converges uniformly on $Y$ over the base $\mathcal{B}_{T}$ to a function $F : Y \rightarrow \mathbb{C}$ and the limit $\lim_{\mathcal{B}_{T}} F_t(y)=A_t$ exists for each $t\in T$, the both repeated limits $\lim_{\mathcal{B}_{Y}}(\lim_{\mathcal{B}_{T}}F_t(y))$ and $\lim_{\mathcal{B}_{T}}(\lim_{\mathcal{B}_{Y}}F_t(y))$ exist and the equality and holds
  $$ 
\lim_{\mathcal{B}_{Y}}(\lim_{\mathcal{B}_{T}}F_t(y))=\lim_{\mathcal{B}_{T}}(\lim_{\mathcal{B}_{Y}}F_t(y)).
$$

This theorem can be found in books of Zorich (Mathematical Analysis II p. 381).
