TITLE: Linear Algebra Versus Functional Analysis
QUESTION [62 upvotes]: As it is mentioned in the answer by Sheldon Axler in this post, we usually  restrict linear algebra to finite dimensional linear spaces and study the infinite dimensional ones in functional analysis. 
I am wondering that what are those parts of the theory in linear algebra that restrict it to finite dimensions. To clarify myself, here is the main question

Question.
  What are the main theorems in linear algebra that are just valid for finite dimensional linear spaces and are propagated in the sequel of theory, used to prove the following theorems?

Please note that I want to know the main theorems that are just valid for finite dimensions not all of them. By main, I mean the minimum number of theorems of this kind such that all other such theorems can be concluded from these.

REPLY [6 votes]: This is mainly a comment for the answer given by avs, but perhaps too long, so I post it as an answer too.
I would add that when $V$ is infinite-dimensional and carrying a topology there are two different notions of the dual vector space for it: 

algebraic dual of $V$ consists of all linear functionals on $V$.
topological dual of $V$ consists of the bounded (continuous) functionals on $V$. 

Both of these duals are usually denoted by $V^*$ and you need to distinguish them from the context. 
The algebraic dual is never isomorphic to $V$ when $V$ is infinite dimensional. Thus we have the following classification:

$V$ is finite-dimensional if and only if it is isomorphic to its
  algebraic dual $V^*$.