Prove that a function $F\in (C([0,1]\times [0,1],\mathbb{R})$ can be uniformly approximately by functions of the form
$$p(x,y)=(a_0+a_1x^2+…+a_{n_{1}}x^{n_{1}})(b_0+b_1y^2+…+b_{n_2}y^{n_2})$$
I'm trying to use Stone-Weierstrass real case for this exercise. But i'm very stuck. Can someone help me?
My attempt:
By Stone Weierstrass i proved that P[0,1] are dense in C[0,1]. As cartesian product of dense set is dense then $P[0,1]\times P[0,1]$ is dense in $C[0,1] \times C[0,1]$.
This implies for each function $f\in F$ exists a polynomic function $p$ such that $||f-p||_\infty<\epsilon$
Let $p(x,y)=(a_0+a_1x^2+…+a_{n_{1}}x^{n_{1}})(b_0+b_1y^2+…+b_{n_2}y^{n_2})$,
Here i'm stuck
Best Answer
From this you already know that $\displaystyle\sum\varphi_{i}\psi_{i}$ approximates $F$, so given $\epsilon\in(0,1)$, we choose polynomials $p_{i},q_{i}$ such that \begin{align*} \|\varphi_{i}-p_{i}\|<\dfrac{\epsilon}{nM},~~~~\|\psi_{i}-q_{i}\|<\dfrac{\epsilon}{nM}, \end{align*} where $M=1+\|\varphi_{1}\|+\cdots+\|\varphi_{n}\|+\|\psi_{1}\|+\cdots+\|\psi_{n}\|$, then \begin{align*} \|\varphi_{i}\psi_{i}-p_{i}q_{i}\|&\leq\|\varphi_{i}-p_{i}\|\|\psi_{i}\|+\|p_{i}\|\|\psi_{i}-q_{i}\|\\ &<\dfrac{\epsilon\|\psi_{i}\|}{nM}+\dfrac{\epsilon(\|\varphi_{i}\|+1)}{nM}\\ &<\dfrac{2\epsilon}{n}, \end{align*} so \begin{align*} \left\|\sum\phi_{i}\psi_{i}-\sum p_{i}q_{i}\right\|\leq\sum\|\phi_{i}\psi_{i}-p_{i}q_{i}\|<2\epsilon, \end{align*} we are done.
If one already recognizes that $C([0,1]\times[0,1])\approx C[0,1]\widehat{\otimes}C[0,1]$, the projective tensor product of copies of $C[0,1]$, the assertion will become immediate.