In this paper, we are interested in giving two characterizations for the so-called property L\(_{o,o}\), a local vector valued Bollobás type theorem. We say that (X, Y) has this property whenever given \(\varepsilon > 0\) and an operador \(T: X \rightarrow Y\), there is \(\eta = \eta (\varepsilon , T)\) such that if x satisfies \(\Vert T(x)\Vert > 1 - \eta \), 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) Bollobás theorem for operators. We prove that the pair (X, Y) has the L\(_{o,o}\) for compact operators if and only if so does \((X, \mathbb {K})\) for linear functionals. This generalizes at once some results due to D. Sain and J. Talponen. Moreover, we present a complete characterization for when \((X \widehat{\otimes }_\pi Y, \mathbb {K})\) satisfies the 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 \((L_p(\mu ) \times L_q(\nu ); \mathbb {K})\) cannot satisfy the L\(_{o,o}\) for bilinear forms.

在本文中，我们有兴趣为所谓的属性L \(_{o,o}\)给出两个特征，这是一个局部向量值 Bollobás 类型定理。我们说 ( X ,  Y ) 在给定\(\varepsilon > 0\)和操作符\(T: X \rightarrow Y\)时具有这个属性，则存在\(\eta = \eta (\varepsilon , T) \)使得如果x满足\(\Vert T(x)\Vert > 1 - \eta \)，则存在\(x_0 \in S_X\)使得\(x_0 \approx x\)和T本身达到它在\(x_0\)处的范数. 这可以看作是运算符的强（尽管局部） Bollobás 定理。我们证明对 ( X ,  Y ) 具有L \(_{o,o}\)用于紧致算子，当且仅当\((X, \mathbb {K})\)用于线性泛函。这立即概括了 D. Sain 和 J. Talponen 的一些结果。此外，我们提出了当\((X \widehat{\otimes }_\pi Y, \mathbb {K})\)满足L \(_{o,o}\)对于严格凸性下的线性泛函或空间之一中的 Kadec-Klee 属性假设。因此，我们概括了与投影张量积的强次可微性相关的文献中的一些结果，并表明\((L_p(\mu ) \times L_q(\nu ); \mathbb {K})\)不能满足双线性形式的L \(_{o,o}\)。