Local theory of integral Banach mapping spaces

In this paper, we mainly investigate the local theory of integral Banach mapping spaces. We introduce notions of integral injectivity, integral local reflexivity, integral exactness and integral nuclearity. We prove that any dual Banach space is integral injective and any Banach space is integral locally reflexive. To our surprise, we also obtain that C is the unique Banach space which is integral exact or integral nuclear.