The key to making a concise mathematical definition of escape velocity is to consider the energy. if an object is launched at its escape velocity, what is the total mechanical energy etotal of the object at a very large (i. e., infinite) distance from the planet? follow the usual convention and take the gravitational potential energy to be zero at very large distances.

If an object is launched at escape velocity, its total mechanical energy at infinite distance is zero.

In fact, the escape velocity is the minimum speed the object should have to escape the gravitational field: this means that when the object approaches infinite, its kinetic energy is the minimum possible, so zero (if it was greater than zero, than it means that the object was launched at larger speed than its escape velocity).

At the same time, the gravitational potential energy of the object at infinite is zero (because hte gravitational potential is taken to be zero at infinite), so the total mechanical energy of the object, which is the sum of kinetic and potential energy, is zero as well.