Projective tensor product

Remarks

The norm is a cross norm: $\pi(x \otimes y) = \norm{x} \, \norm{y}$. The bound $\le$ comes from the representation with one term. For $\ge$, take functionals $f \in X^*$ and $g \in Y^*$ of norm $1$ with $f(x) = \norm{x}$ and $g(y) = \norm{y}$, given by the Hahn–Banach theorem. For any representation, $\norm{x} \, \norm{y} = \abs{\sum_i f(x_i) g(y_i)} \le \sum_i \norm{x_i} \, \norm{y_i}$.

The infimum is needed: $u = 0$ can be written as $x \otimes y + x \otimes (-y)$, with cost $2 \norm{x} \, \norm{y}$. A single representation does not give a well-defined number.

Universal property: bounded bilinear maps $B \colon X \times Y \to Z$ into a Banach space $Z$ correspond to bounded linear maps $X \widehat{\otimes}_\pi Y \to Z$ with the same norm. In particular, the dual of $X \widehat{\otimes}_\pi Y$ is the space of bounded linear maps from $X$ to $Y^*$. For example, $\ell^1 \widehat{\otimes}_\pi \ell^1$ is isometric to $\ell^1(\N \times \N)$.