  In this paper, we are interested in giving two characterizations for the
so-called property {\bf L}$_{o,o}$, a local vector valued Bollob\'as type
theorem. We say that $(X, Y)$ has this property whenever given $\eps > 0$ and
an operador $T: X \rightarrow Y$, there is $\eta = \eta(\eps, T)$ such that if
$x$ satisfies $\|T(x)\| > 1 - \eta$, then there exists $x_0 \in S_X$ such that
$x_0 \approx x$ and $T$ itself attains its norm at $x_0$. This can be seen as a
strong (although local) Bollob\'as theorem for operators. We prove that the
pair $(X, Y)$ has the {\bf L}$_{o,o}$ for compact operators if and only if so
does $(X, \K)$ for linear functionals. This generalizes at once some results
due to D. Sain and J. Talponen. Moreover, we present a complete
characterization for when $(X \pten Y, \K)$ satisfies the {\bf L}$_{o,o}$ for
linear functionals under strict convexity or Kadec-Klee property assumptions in
one of the spaces. As a consequence, we generalize some results in the
literature related to the strongly subdifferentiability of the projective
tensor product and show that $(L_p(\mu) \times L_q(\nu); \K)$ cannot satisfy
the {\bf L}$_{o,o}$ for bilinear forms.
