Some companions of the Grüss inequality in inner product spaces.
We present some convergence theorems for the HK-integral of functions taking values in a locally convex space. These theorems are based on the concept of HK-equiintegrability.
The main result is as follows. Let X be a Banach space and let Y be a closed subspace of X. Assume that the pair has the λ-bounded approximation property. Then there exists a net of finite-rank operators on X such that and for all α, and and converge pointwise to the identity operators on X and X*, respectively. This means that the pair (X,Y) has the λ-bounded duality approximation property.