A topological vector space is a mathematical structure that combines the concepts of vector spaces and topology. In this space, vectors can be added together and multiplied by scalars, while also having a topology that allows for the notion of continuity. This means that we can discuss concepts like convergence and limits in a way that respects the vector space operations. The topology on a topological vector space is typically defined by a collection of open sets, which helps in understanding the space's structure. Common examples include Banach spaces and Hilbert spaces, which are both important in functional analysis and have applications in various fields such as physics and engineering.