The following question is from Banach Algebra Techniques in Operator Theory written by Ronald G. Douglas.
Assume both $X, Y$ are Banach spaces and $X \otimes Y$ is the algebraic tensor product. Let ${X^*}_{\leq1}$ be the closed unit ball in $X^*$. For $w \in X \otimes Y$, define $\|w\|_i = \sup\{\left|\sum_{k = 1}^n \phi(x_k) \psi(y_k)\right| : x_k \in X, y_k \in Y, w = \sum_{k = 1}^n x_k \otimes y_k\}$ (one of $w$'s expression in $X \otimes Y$), $\phi \in {X^*}_{\leq1}, \psi \in {Y^*}_{\leq1}$}. One can check this is a norm on $X \otimes Y$ and we let ($X \mathbin{\bar{\otimes}} Y, \| \cdot \|_i$) be the completion of the set ($X \otimes Y, \| \cdot \|_i$).
Now assume $X, Y$ are both compact Hausdorff topological spaces and hence ($C(X), \| \cdot \|_{\infty}$), ($C(Y), \| \cdot \|_{\infty}$) are Banach spaces. Show that ($C(X) \bar{\otimes} C(Y), \| \cdot \|_i$) is isometrically isomorphic to ($C(X \times Y), \| \cdot \|_{\infty}$). Here $X \times Y$ is equipped with the product topology.
Note that any norm $\|\cdot\|$ in $C(X) \oplus C(Y)$ (the direct sum of two Banach Spaces) is equivalent to $\|\cdot\|_1$ because both $C(X), C(Y)$ are Banach spaces equipped with $\|\cdot\|_{\infty}$ (hence $\|(f_x, f_y)\|_1 = \|f_x\|_{\infty} + \|f_y\|_{\infty}$. Meanwhile, one can find a homeomorphism between $C(X)\oplus C(Y)$ and $C(X\times Y)$ because $\|f\|_{\infty} \leq \|f_x\|_{\infty} + \|f_y\|_{\infty} \leq 2\|f\|_{\infty}$. Hence I directly start finding relation between ($C(X)\oplus C(Y), \|\cdot\|_1$) and ($C(X) \mathbin{\bar{\otimes}} C(Y), \|\cdot\|_i$)
$$\Large Question Part$$
Say $w \in C(X)\bar{\otimes} C(Y)$ and here I have difficulty finding upper bound of $\|w\|_i$ with respect to $\|\cdot\|_1$. Naively I consider partition of unity of $X$, say {$P_i, i \leq n$} and $\sum_{i \leq n}fP_i$ is one to break down $f$. Hence this could be one of the expression of $f$ part in $w$. I do not know if $n$ is the max number of pieces of $f$ I can break down.
According to hints in the book, by Krein-Milman, it suffices to consider extreme points in $X^*$ and $Y^*$. Before using this, I believe I need to collect enough information of $w$.
