Well, to be fair, functional analysis is not part of linear algebra proper. (If you want to get more abstract, you go to rings and modules and from there to category theory.)
Typically, linear algebra is understood to be the study of finite-dimensional vector spaces, so functional analysis is not necessarily part of it.
However, things like the vector space of polynomials of degree at most n, the vector space of all homomorphisms between two vector spaces, the dual space of a vector space, etc. are all concepts that belong to linear algebra proper yet are more "abstract" than just "computations with matrices".
