Alternative Definitions of the Projective Tensor Norm on X⊗Y

# Alternative Definitions of the Projective Tensor Norm on X⊗Y

Recall from The Projective Tensor Product of X⊗Y page that if $X$ and $Y$ are normed linear spaces then we defined the projective tensor norm on $X \otimes Y$ for every $u \in X \otimes Y$ by:

(1)
\begin{align} \quad p(u) = \inf \left \{ \sum_{i=1}^{m} \| x_i \| \| y_i \| : u = \sum_{i=1}^{m} x_i \otimes y_i \right \} \end{align}

We define the projective tensor product of the normed spaces $X$ and $Y$ to be the completion of $X \otimes Y$ with respect to the projective tensor norm $p$, and denoted it by $X \otimes_p Y$. Moreover, we noted that every $u \in X \otimes_p Y$ can be written in the form $\displaystyle{u = \sum_{i=1}^{\infty} x_i \otimes y_i}$ with $\displaystyle{\sum_{i=1}^{\infty} \| x_i \| \| y_i \| < \infty}$ and that:

(2)
\begin{align} \quad p(u) = \inf \left \{ \sum_{i=1}^{\infty} \| x_i \| \| y_i \| : u = \sum_{i=1}^{\infty} x_i \otimes y_i \: \mathrm{and} \: \sum_{i=1}^{\infty} \| x_i \| \| y_i \| < \infty \right \} \end{align}

Sometimes it is convenient to have alternative formulas for the projective tensor norm $p$. The following proposition gives us some of these alternative formulas.

 Proposition 1: Let $X$ and $Y$ be normed linear spaces and let $X \otimes_p Y$ be the projective tensor product of $X$ and $Y$. Then for all $u \in X \otimes_p Y$ we have that: $\displaystyle{p(u) = \inf \left \{ \sum_{i=1}^{\infty} |\lambda_i| : u = \sum_{i=1}^{\infty} \lambda_ix_i \otimes y_i, \: \sum_{i=1}^{\infty} |\lambda_i| < \infty, \: \| x_i \| = \| y_i \| = 1 \right \} = \inf \left \{ \sum_{i=1}^{\infty} |\lambda_i|\| x_i \| \| y_i \| : u = \sum_{i=1}^{\infty} \lambda_i x_i \otimes y_i, \: \sum_{i=1}^{\infty} |\lambda_i| < \infty, \: x_i, y_i \to 0 \right \}}$.

The following proposition gives us a formula to compute $p(u)$ when $u$ is just an element of $X \otimes Y$ (this formula is not generally valid for tensors in the completion $X \otimes_p Y$). It tells us that $p(u)$ can be computed by computing the projective tensor norm of $u$ in tensor products of finite-dimensional subspaces of $X$ and $Y$ containing $u$.

 Proposition 2: Let $X$ and $Y$ be normed linear spaces and let $(X \otimes Y, p)$ be the tensor product $X \otimes Y$ with the norm $p$ (i.e., the non-completed tensor product with norm $p$). Then for all $u \in X \otimes Y$ we have that $p(u, X \otimes Y) = \inf \{ p(u, M \otimes N) : u \in M \otimes N \: \mathrm{and} \: \mathrm{dim} (M), \mathrm{dim}(N) < \infty \}$.

$p(u, X \otimes Y)$ denotes the projective tensor norm of $u$ in $X \otimes Y$ while $p(u, M \otimes N)$ denotes the projective tensor norm of $u$ in $M \otimes N$. Note that if $M$ is a subspace of $X$ and $N$ is a subspace of $Y$ we have that if $u \in M \otimes N$ then every representation of $u$ as a finite sum of tensors $m_i \otimes n_i$ in $M \otimes N$ is also a representation of $u$ in $X \otimes Y$. Thus $p(u, X \otimes Y) \leq p(u, M \otimes N)$.

• Proof: Let $X$ and $Y$ be normed linear spaces and let $u \in X \otimes Y$. By the definition of $p(u)$ being an infimum, for all $\epsilon > 0$ there exists a representation $u = \sum_{i=1}^{n} x_i \otimes y_i$ such that:
(3)
\begin{align} \quad \sum_{i=1}^{n} \| x_i \| \| y_i \| \leq p(u, X \otimes Y) + \epsilon \quad (*) \end{align}
• Consider the sets $\{ x_1, x_2, ..., x_n \}$ of $X$ and $\{ y_1, y_2, ..., y_n \}$ of $Y$ and let:
(4)
\begin{align} \quad M = \mathrm{span} (x_1, x_2, ..., x_n) \\ \quad N = \mathrm{span} (y_1, y_2, ..., y_n) \end{align}
• Then $M$ is a finite-dimensional subspace of $X$ and $N$ is a finite-dimensional subspace of $Y$. Moreover, $u \in M \otimes N$. Therefore $p(u, M \otimes N) \leq \sum_{i=1}^{n} \| x_i \| \| y_i \|$. Combining this with the inequality at $(*)$ gives us that:
(5)
\begin{align} \quad p(u, M \otimes N) \leq p(u, X \otimes Y) + \epsilon \end{align}
• Since this holds true for all $\epsilon > 0$ we see that $p(u, M \otimes N) \leq p(u, X \otimes Y)$ for all finite-dimensional subspaces $M$ of $X$ and $N$ of $Y$. Combining this with the fact that $p(u, X \otimes Y) \leq p(u, M \otimes N)$ for any arbitrary subspaces $M$ of $X$ and $N$ of $Y$ gives us that:
(6)
\begin{align} \quad p(u, X \otimes Y) = \inf \{ p(u, M \otimes N) : u \in M \otimes N \: \mathrm{and} \: \mathrm{dim}(M), \mathrm{dim}(N) < \infty \} \quad \blacksquare \end{align}