Semi‐normed Spaces

This chapter defines normed spaces and separated semi-normed spaces. A semi-normed space is a vector space endowed with a non-empty family of seminorms. The chapter proves that every normed space is a separated semi-normed space. A vector space endowed with several norms is a separated seminormed space but is not a normed space. The chapter concerns the study of the separated semi-normed spaces, where they play a crucial role in the study of PDEs. Semi-normed space is equivalent to a locally convex topological vector space according to von Neumann's theorem. The construction with semi-norms has three advantages (the first is decisive) compared to the construction with a locally convex topology: effectiveness, simplicity, and numerical values. The chapter presents a few calculation rules and shows that any semi-norm is contracting. It then defines convergent sequences and Cauchy sequences of a separated seminormed space.