  We study different aspects of the connections between local theory of Banach
spaces and the problem of the extension of bilinear forms from subspaces of
Banach spaces. Among other results, we prove that if $X$ is not a Hilbert space
then one may find a subspace of $X$ for which there is no Aron-Berner
extension. We also obtain that the extension of bilinear forms from all the
subspaces of a given $X$ forces such $X$ to contain no uniform copies of
$\ell_p^n$ for $p\in[1,2)$. In particular, $X$ must have type $2-\epsilon$ for
every $\epsilon>0$. Also, we show that the bilinear version of the
Lindenstrauss-Pe{\l}czy\'nski and Johnson-Zippin theorems fail. We will then
consider the notion of locally $\alpha$-complemented subspace for a reasonable
tensor norm $\alpha$, and study the connections between $\alpha$-local
complementation and the extendability of $\alpha^*$ -integral operators.
