A quantitative version of the Kolmogorov–Riesz theorem

Abstract

In the present paper, we prove a generalization of the celebrated Kolmogorov–Riesz theorem in the $$L^{p}([0,1],X)$$ spaces, with $$1\le p <\infty$$ and X a Banach space. Specifically, our result is a quantitative version of such theorem. Our main tool is the so-called degree of nondensifiability, which measures (in the specified sense), the distance from a given convex subset of a Banach space to the class of its Peano Continua.