TOPOLOGICAL VECTOR SPACE AND ITS PROPERTIES

The main aim of this project is to learn a branch of Mathematics that studies vector spaces endowed with some kind of limit-related structure (e.g. inner product, norm, topology, etc.) and the linear functions defined on these spaces and respecting these structures in a suitable sense. Specifically we will learn vector space with some topology on it (called topological vector space). A topological vector space (also called a linear topological space) is one of the basic structures investigated in functional analysis. A topological vector space is a vector space (an algebraic structure) which is also a topological space, thereby admitting a notion of continuity. More specially, its topological space has a uniform topological structure, allowing a notion of uniform convergence.