About Error Bounds in Metrizable Topological Vector Spaces

This paper aims to present some sufficient criteria under which a given function between two spaces that are either topological vector spaces whose topologies are generated by metrics or metrizable subsets of some topological vector spaces, satisfies the error bound property. Then, we discuss the Hoffman estimation and obtain some results for the estimate of the distance to the set of solutions to a system of linear equalities. The advantage of our estimate is that it allows to calculate the coefficient of the error bound. The applications of this presentation are illustrated by some examples.