Approximation of Inductive Tensor Product $C(X) \bar{\otimes} C(Y)$

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 $$w$$.

• Regarding terminology: I think this is usually called the injective tensor product of Banach spaces, not the "inductive" tensor product – Yemon Choi Jan 21 at 4:20
• 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. – Nik Weaver Jan 21 at 5:05
• 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. – Nik Weaver Jan 21 at 11:52
• You are wrong. Consider the case where $X$ and $Y$ are both singletons. – Nik Weaver Jan 22 at 17:45
• No ... no. I really think you should start by trying to understand the finite dimensional case. – Nik Weaver Jan 23 at 23:59