@StanShunpike I *think* this answers your question
"Mathematically, Heisenberg theory uses the space l2, while Schrodinger theory uses the space L2. A consequence of the Riesz-Fisher theorem is that the spaces l2 and L2 are isomorphic, a result proved in this book. The two theories leads to the same physical results, and in consequence are equivalent, although different in the mathematical content ! You need to buy this book ! "
http://www.amazon.com/Elements-Functions-Functional-Analysis-Mathematics/dp/0486406830/