In this short article, we shall study one-dimensional local Dirichlet spaces associated with linear diffusions.One result, which has its independent interest, is to prove that irreducibility implies the uniqueness of symmetrizing measure for fight Markov processes.The first result is to give a representation for any 1-dim local, irreducible and regular Dirichlet space and the second result is a necessary and sufficient condition for a Dirichlet space to be regular subspace of another Dirichlet space.