Functional

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Let $\mu$ be a $\sigma$-finite measure on a measure space $M$ and net $\nu$ be a $\sigma$-finite measure on a measure space $N$. Let:

(1)
\begin{align} \quad L^1(M, \mu) &= \left \{ f : M \to \mathbb{C} : \int_M |f| \: d \mu < \infty \right \} \\ \quad L^1(N, \nu) &= \left \{ f : N \to \mathbb{C} : \int_N |f| \: d \nu < \infty \right \} \end{align}

Let $\mu \times \nu$ be the product measure of $\mu$ and $\nu$ on $M \times N$ (this is unique since $\mu$ and $\nu$ are $\sigma$-finite) and consider the space $L^1(M \times N, \mu \times n)$ as well.

Let $T : L^1(M, \mu) \times L^1(N, \nu) \to L^1(M \times N, \mu \times \nu)$ be defined for all $f \in L^1(M, \mu)$ and all $g \in L^1(N, \nu)$ by $T(f, g) = fg$, where $fg : L^1(M \times N, \mu \times \nu)$ is defined for all $(m, n) \in M \times N$ by $(fg)(m, n) = f(m)g(n)$.

Since $T : L^1(M, \mu) \times L^1(N, \nu)$ is a bilinear map, by The Existence of a Linear Map σ on X⊗Y to Z that Matches a Bilinear Map on X×Y to Z there exists a linear map $\sigma : L^1(M, \mu) \otimes L^1(N, \nu) \to L^1(M \times N, \mu \times \nu)$ such that:

(2)
\begin{align} \quad \sigma(f \otimes g) = T(f, g) \end{align}

That is, for all $(m, n) \in M \times N$:

(3)
\begin{align} \quad \sigma(f \otimes g)(m, n) &= T(f, g)(m, n) \\ &= (fg)(m, n) \\ &= f(m)g(n) \end{align}

Let $u \in L^1(M, \mu) \otimes L^1(N, \nu)$ and write $u = \sum_{i} f_i \otimes g_i$. Then observe that:

(4)
\begin{align} \quad \| \sigma (u) \|_1 &= \left \| \sigma \left ( \sum_{i} f_i \otimes g_i \right ) \right \|_1 \\ &= \left \| \sum_{i} \sigma (f_i \otimes g_i) \right \|_1 \\ &= \left | \sum_{i} T(f_i, g_i) \right \|_1 \\ &\leq \sum_{i} \int_{M \times N} |f_ig_i| \: d (\mu \times \nu) \\ &\leq \sum_{i} \int_M |f_i| d \: mu \int_N |g_i| \: d \nu \\ &\leq \sum_{i} \| f_i \|_1 \| g_i \|_1 \\ \end{align}

Since the above equality holds true for all representations of $u$ we have that $\| \sigma (u) \|_1 \leq p(u)$ for all $u \in L^1(M, \mu) \otimes L^1(N, \nu)$. So the linear map $\sigma : L^1(M, mu) \otimes L^1(N, \nu) \to L^1(M \times N, \mu \times \nu)$ be be extended to a linear map $\sigma : L^1(M, \mu) \otimes_p L^1(N, \nu) \to L^1(M \times N, \mu \times \nu)$.