Let be a topological vector space (TVS) over , that is, is a vector space equipped with a topology so that vector addition and scalar multiplication are continuous. We call the topology that starts with the original, starting, or given topology (the reader is cautioned against using the terms "initial topology" and "strong topology" to refer to the original topology since these already have well-known meanings, so using them may cause confusion). We may define a possibly different topology on using the topological or continuous dual space , which consists of all linear functionals from into the base field that are continuous with respect to the given topology.
Let be a topological vector space (TVS) over , that is, is a vector space equipped with a topology so that vector addition and scalar multiplication are continuous. We call the topology that starts with the original, starting, or given topology (the reader is cautioned against using the terms "initial topology" and "strong topology" to refer to the original topology since these already have well-known meanings, so using them may cause confusion). We may define a possibly different topology on using the topological or continuous dual space , which consists of all linear functionals from into the base field that are continuous with respect to the given topology