## Introduction to Dual Spaces

 When you have a finite-dimensional vector space V (in quantum mechanics they are, as a rule, complex spaces), then the space of all linear complex valued functions (linear functionals) on V is of the same dimension as V. It is called the dual of V. But when V is infinite dimensional, then you may want your functionals to be continuous in some topology. This way you get the topological dual. The weaker continuity conditions you impose on your functionals (always linear), the more functionals are included in the dual. In QM we take for V some dense vector subspace of the Hilbert space H, impose some appropriate topology, and then build the dual V*. Normally we get natural embeddings: $$V\subset H\subset V^*$$ This is how Gelfand's triples come into live. You "momentum eigenstates" can be then considered as vectors in V*. But V* is no longer a Hilbert space. Therefore scalar products between momentum eigenstates are not defined as finite "numbers".
 if you have a vector space $V$, then the dual space $V^*$ is just the set of all real-valued linear functions on $V$ (commonly called linear functionals). This set is a vector space itself, and they mention isomorphisms because it has the same dimension as the original vector space $V$, which for finite-dimensional vector spaces makes them isomorphic. so, the dual space is just all the functions that take a vector in your vector space as input and then return a number.