Topological Groups and Topological Vector Spaces

This chapter studies certain topological-algebraic structures such as topological groups and topological vector spaces. The book Basic Topology, Volume 1 of the present series of books studies the general properties of topological spaces and their continuous maps. But this chapter studies the topological spaces with other structures (algebraic) compatible with the given topological structures. For example, the circle group $$ S^1$$ S 1 in the complex plane $$ \textbf{C},$$ C , the 3-spheres $$ S^3$$ S 3 (group of unit quaternions), the general linear group $$ GL ( n. \textbf{R})$$ G L ( n . R ) , $$ GL ( n. \textbf{C})$$ G L ( n . C ) , etc., admit a natural group structure under usual multiplication such that their usual topological and algebraic group structures are compatible in the sense that the corresponding group operations are continuous. It asserts that the concept of a topological group is precisely that concept in which the algebraic and topological structures are united and interrelated. This phenomenon leads to the concept of topological groups.