Topological vector space

Summary

A topological vector space is a vector space over $\R$ or $\C$ with a topology in which addition and scalar multiplication are continuous. It is the common frame of functional analysis: normed spaces, Banach spaces, Hilbert spaces and locally convex spaces are all examples. Under addition it is a commutative topological group, so its topology is determined by the neighbourhoods of $0$. Continuity of linear maps, convergence of series and duality are all studied in this setting.