On uniform convexity of linear metric spaces

UDC 517 The notion of uniform convexity was introduced and discussed in normed linear spaces by Clarkson [Trans. Amer. Math. Soc., 40, 396–414 (1936)]. Later, this idea attracted attention of other researchers who studied these spaces in depth and also introduced and analyzed in detail numerous other weaker forms of uniform convexity. Further, the idea of uniform convexity and its weaker forms was also extended to linear metric spaces. Over the years, the researchers investigated and analyzed the properties of linear metric spaces equipped either with uniform convexity or with its weaker forms, and studied their relationships, inheritance of these properties by quotient spaces, and the applications of these properties to many other domains of mathematics. In this article, we survey the available literature on these topics in linear metric spaces.