# Approximation of a two-variable function by tensor products

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

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