Quantum theory differs dramatically in that it replaces physical observables such as the position of the system, the time at which that observation is made, the mass, and the velocity of the system at the instant of observation with the notion of operator observables. Operators as observables change the notion of what is measurable and brings to the table the unavoidable conclusion of the Max Born probability theory. In this framework of nondeterminism, the probability of finding the system in a particular observable state is given by a dynamic probability density that is defined as the absolute value squared of the solution to the Schrodinger equation. The fact that probability densities are integrable and normalizable to unity imply that the solutions to the Schrodinger equation must be square integrable. The vector space of infinite sequences, whose square summed up is a convergent series, is known as ℓ ℓ 2 (pronounced "little ell two"). It is in one-to-one correspondence with the infinite dimensional vector space of square-integrable functions, L 2 ( R d ) , from the Euclidean space R d to the complex plane, C . For this reason, ℓ ℓ 2 and L 2 ( R d ) are often referred to indiscriminately as "the" Hilbert space. This is rather misleading because R d is also a Hilbert space when equipped and completed under the Euclidean inner product, albeit a finite dimensional space.