Embedding singular linear system with initial condition into distribution space

For inconsistent initial value problem of singular linear system, passing from classical solution to distributional solution is formulated as a process of embedding the system with initial condition into distribution space. Three technical notions, function space on R+ with given initial value, consistent extension of the pair of the function space and input, and order exchange of the two operations of embedding and differentiating, result in a rigorous definition of distributional solution.