Norm-Attaining Tensors and Nuclear Operators

Given two Banach spaces X and Y, we introduce study a concept of norm-attainment in the space nuclear operators $$\mathcal N(X,Y)$$ projective tensor product $$X\widehat{\otimes }_\pi Y$$ . We exhibit positive negative examples where both previous hold. also problem whether class elements which attain their norms is dense or not. prove that, for concepts, density norm-attaining holds large Y which, particular, covers all classical spaces. Nevertheless, present failing approximation property such way that not dense. discuss some relations applications our work to theory throughout paper.