Iterative spaces of locally integrable functions

Iterative spaces of locally integrable functions are found. The set of locally integrable generally complex-valued functions satisfying certain conditions are considered for an interval. The embedding for these spaces are found to be everywhere dense. This result shows that all norms of this function are equivalent and characterize the same classes as the spaces. A domain that is the class of vector-valued functions with values in a Banach space is defined. It is proved by applying Hille-Phillips assertion that these classes are Banach spaces with the norm.