Abstract
This chapter focuses on the metrization of groups and vector spaces. Relations among several types of spaces are depicted by a chart in the chapter. A topological vector space (TVS) represents the vector space equipped with a topology that makes the vector space operations continuous. It also informs about topological Abelian group (TAG) that is an Abelian group equipped with a topology that makes the group operations continuous. This makes the topology translation-invariant, so there is a uniform structure naturally associated with that topology. Even when scalars are present (i.e., in a TVS), the most basic properties of the uniform structure do not involve the scalars; therefore it is natural to first introduce uniform structure in the simpler and more general setting of TAGs. In addition an F-space is a vector space topologized by an F-norm that is complete. Equivalently, it is a complete metrizable topological vector space.
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