I am reading the book Introduction to tensor products of Banach spaces by Ryan. In chapter 2, page 15, author said to define a norm on $X\otimes Y=$ span $\{x\otimes y: x \in X, y \in Y\}$ , it is natural to require that $\|x\otimes y\| \leq \|x\| \|y\|$ for all x in X and y in Y.
I don't understand why we need this requirement. May be it is obvious but it is not so clear to me. I am new in this area. Any hints or suggestions would be appreciated. Thank you very much.
$x\otimes y$ is a linear map from $B(X\times Y)$ to $\mathbb{R}$ such that $(x\otimes y) (A) = A(x, y) $
$B(X\times Y)$ is space of all linear maps from $X\times Y$ to $\mathbb{R}$