TITLE: Why study finite-dimensional vector spaces in the abstract if they are all isomorphic to $R^n$?
QUESTION [48 upvotes]: Timothy Gowers asks Why study finite-dimensional vector spaces in the abstract if they are all isomorphic to $R^n$? and lists some reasons. The most powerful of these is probably 

There are many important examples throughout mathematics of infinite-dimensional vector spaces. If one has understood finite-dimensional spaces in a coordinate-free way, then the relevant part of the theory carries over easily. If one has not, then it doesn't.

I mean sure, but what else? Does anyone know examples of specific vector spaces?

REPLY [6 votes]: Some examples of finite-dimensional vector spaces that are useful in my field of practice, convex optimization:

real or complex scalars (yes, $\mathbb{R}^1/\mathbb{C}^1$)
$\mathbb{C}^n$, the space of complex vectors
the space of $m\times n$ real or complex matrices
$\mathcal{S}^n$, the space of $n\times n$ symmetric matrices
$\mathcal{H}^n$, the space of $n\times n$ Hermitian matrices
other spaces of matrices with specific structure; e.g., Toeplitz matrices, Hankel matrices
Cartesian products of two or more of the above. For example, a tuple consisting of a symmetric matrix, a vector, and a scalar.

Knowing that these vector spaces are isomorphic to $\mathbb{R}^n$ simplifies the study of convex analysis and convex optimization tremendously, because most of results indeed deriving only once---with appeals to the isomorphism used to quickly apply those results to arbitrary finite-dimensional spaces.
How inconvenient would it be for us to avoid discussion of these other vector spaces? Well, let's consider a simple example. An important function in semidefinite programming is the maximum eigenvalue function on symmetric matrices:
$$f:\mathcal{S}^n\rightarrow\mathbb{R}, \qquad f(X) = \lambda_{\max}(X) = \max_{v\neq 0} (v^TXv)/(v^Tv)$$
This a convex function of its input $X$, and as a result we can build practically useful and tractable optimization problems that incorporate it into objective functions or constraints. (Other matrix functions, such as the minimum eigenvalue, the logarithm of the determinant, etc., arise frequently as well.)
Now, because of the isomorphism, we know that there is another function $g:\mathbb{R}^{n(n+1)/2}\rightarrow\mathbb{R}$ that could be used to represent $f$. In other words, to compute $g(x)$, we'd take the $n(n+1)/2$ values of $x$, arrange them in the lower triangle of a matrix, copy the off-diagonal elements to the upper triangle to symmetrize things, then compute the eigenvalue of that matrix. 
Whew! Do I really have to do that every time I want to work with a function defined on a non-$\mathbb{R}^n$ vector space? I hope not. Sure, sometimes it might be easier to prove a particular technical result if I do; but no, most of the time I want to work in the natural vector space the problem suggests. Knowing how the principles of vector spaces apply to spaces beyond $\mathbb{R}^n$ enables me to avoid constant translation to and fro.