Let $X$ and $Y$ be compact metric spaces and $f: X \times Y \to \mathbb{R}$ be a continuous function.

We know that, for every $n \in \mathbb{N}$, by the Stone-Weierstrass theorem, there exist $k_n \in \mathbb{N}$ and continuous functions $f^{(1,n)}_i$, $i \in \{1, \ldots, k_n\}$, on $X$ and continuous functions $f^{(2,n)}_i$, $i \in \{1, \ldots, k_n\}$, on $Y$, such that

$$ \sup_{x \in X, y \in Y } \bigg| f(x,y) - \sum_{i=1}^{k_n} f^{(1,n)}_i(x) f^{(2,n)}_i(y) \bigg| < \frac{1}{n} .$$

Is it possible to choose these functions such that

$$ \sup_{n \in \mathbb{N}} k_n < + \infty, $$ $$ \sup_{n \in \mathbb{N}} \sup_{i \in \{1, \ldots, k_n\}}\sup_{x \in X} \Big| f^{(1,n)}_i(x) \Big| < +\infty, \quad \quad \sup_{n \in \mathbb{N}} \sup_{i \in \{1, \ldots, k_n\}}\sup_{y \in Y} \Big| f^{(2,n)}_i(y) \Big| < +\infty \quad ? $$