I have a question about the usual definition of a Sobolev spaces.
Especially, let $f \in W^{1,2} (\Bbb R^n)$. I have thought that this notation means that $ \int_{\Bbb R^n} |f|^2 + | \partial_1 f|^2 + \cdots + | \partial_n f|^2 < \infty $ where $\partial_i$ means that $i$th weak partial derivative. But I am wondering that this notation may include the product of derivatives, that is, $ \int_{\Bbb R^n} |f|^2 + | \partial_1 f|^2 + \cdots + |\partial_n f |^2 + |\partial_1 f | | \partial_2 f | + \cdots + | \partial_{n-1} f | | \partial_n f| < \infty.$
Which one is true in general?