He introduced a new type of integral, the Hellinger integral in his doctoral thesis and, jointly with Hilbert, he produced the important Hilbert -Hellinger theory of forms.
Siya ang nagpasimula ng isang bagong uri ng mahalaga, ang Hellinger mahalaga sa kanyang doktor at sanaysay, sama-sama sa Hilbert, siya ginawa ang mahalagang Hilbert-Hellinger teorya ng form.