**I still cannot grasp the difference between Hilbert space and Euclidian space. Do you have a better explanation for it?**

A Euclidean space is a special subset of a Hilbert space. They are both vector spaces, which are defined the same way, and the vectors in these spaces follow the same rules. However, the vectors themselves in a Euclidean space can only involve real numbers and can only have a finite number of dimensions. On the other hand, the vectors themselves in a Hilbert space can, in general, involve complex numbers and can, in general, have an infinite number of dimensions. The last two differences make Hilbert spaces general enough to be used in Quantum Mechanics.

**Where will we use Hilbert spaces?**

A generalized vector space such as a Hilbert space can represent a possibly complex-valued function over a specific domain. It is often useful, for a variety of reasons, to represent that function in terms of a linear combination of eigenfunctions. This type of analysis is useful for systematically approximating the function to get a big picture understanding of the behavior of the system that is being mathematically represented by the function. Concrete examples will be discussed throughout the semester.