In this paper, we study almost derivations on Banach algebras. These are linear maps which satisfy the derivation identity within some nonnegative scalar \(\epsilon \) . We also consider the situation where an almost derivation is almost inner, that is within \(\delta \) of some inner derivation. This leads to the quantitative notions of \((\epsilon ,\delta )\) -amenability and \((\epsilon ,\delta )\) -weak amenability for Banach algebras. Similarly, we define the notions of \((\epsilon ,\delta )\) -contractiblity, \(\epsilon \) -biflatness, and \(\epsilon \) -biprojectivity for Banach algebras. We give concrete examples showing that these relaxations are strictly weaker than their classical counterparts. We prove quantitative analogs of the known relations between cohomological properties of Banach algebras.
