ABSTRACT
In this chapter, we present a brief survey of the Steinhaus type theorems over complete, non-trivially valued fields proved so far. By a complete, nontrivially valued K, we mean K = R https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781351142120/c81116b1-56b8-4322-b026-4912085dda0d/content/eq8124.tif"/> (the field of real numbers) or C https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781351142120/c81116b1-56b8-4322-b026-4912085dda0d/content/eq8125.tif"/> (the field of complex numbers) or a complete, non-trivially valued, non-archimedean field. For classical summability theory, standard references are Cooke (1950), Hardy (1949), Maddox (1970), while, for summability theory over non-archimedean fields, a standard reference is Natarajan (2015). Basic knowledge of Analysis - both classical and non-archimdean - is assumed, standard references for non- archimedean analysis being Bachman (1964), Narici et al. (1971). With this very brief introduction, we shall recall some basic concepts and theorems in summability theory.
