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 $w$.

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    $\begingroup$ Regarding terminology: I think this is usually called the injective tensor product of Banach spaces, not the "inductive" tensor product $\endgroup$ – Yemon Choi Jan 21 at 4:20
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    $\begingroup$ What is $C(X) \times C(Y)$? The cartesian product? If so, it can't be homeomorphic to $C(X \times Y)$. Look at the case where $X$ and $Y$ are finite, the dimensions don't match. $\endgroup$ – Nik Weaver Jan 21 at 5:05
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    $\begingroup$ The vector space dimension of $C(X) \oplus C(Y)$ is the sum of $|X|$ and $|Y|$, the dimension of $C(X\times Y)$ is their product. $\endgroup$ – Nik Weaver Jan 21 at 11:52
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    $\begingroup$ You are wrong. Consider the case where $X$ and $Y$ are both singletons. $\endgroup$ – Nik Weaver Jan 22 at 17:45
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    $\begingroup$ No ... no. I really think you should start by trying to understand the finite dimensional case. $\endgroup$ – Nik Weaver Jan 23 at 23:59

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