Most textbooks say, that given a (countable) basis ${|\phi_n\rangle}$ of a Hilbert space, that every vector $|\psi\rangle$ of the space can be written as:

$$\psi\rangle=\sum_{n=1}^\infty a_n|\phi_n\rangle$$

But this an infinite linear combination, and every vector should be expressed by a finite linear combination.

So I think that they are an orthogonal set, which generates the vector space as an infinite linear combination (the difference between infinite and finite LC means that orthonormal sets are usually smaller than basis).

But for the rest of the properties, ${|\phi_n\rangle}$ looks like a basis to me.

So, is ${|\phi_n\rangle}$ a basis or just an orthogonal set?