AbstractIn this paper, we are interested in giving two characterizations for the so-called property L$$_{o,o}$$
o
,
o
, a local vector valued Bollobás type theorem. We say that (X, Y) has this property whenever given $$\varepsilon > 0$$
ε
>
0
and an operador $$T: X \rightarrow Y$$
T
:
X
→
Y
, there is $$\eta = \eta (\varepsilon , T)$$
η
=
η
(
ε
,
T
)
such that if x satisfies $$\Vert T(x)\Vert > 1 - \eta $$
‖
T
(
x
)
‖
>
1
-
η
, then there exists $$x_0 \in S_X$$
x
0
∈
S
X
such that $$x_0 \approx x$$
x
0
≈
x
and T itself attains its norm at $$x_0$$
x
0
. This can be seen as a strong (although local) Bollobás theorem for operators. We prove that the pair (X, Y) has the L$$_{o,o}$$
o
,
o
for compact operators if and only if so does $$(X, \mathbb {K})$$
(
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 $$(X \widehat{\otimes }_\pi Y, \mathbb {K})$$
(
X
⊗
^
π
Y
,
K
)
satisfies the L$$_{o,o}$$
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 $$(L_p(\mu ) \times L_q(\nu ); \mathbb {K})$$
(
L
p
(
μ
)
×
L
q
(
ν
)
;
K
)
cannot satisfy the L$$_{o,o}$$
o
,
o
for bilinear forms.
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