Why study finite-dimensional vector spaces in the abstract if they are all isomorphic to RnR^n?

Timothy Gowers asks Why study finite-dimensional vector spaces in the abstract if they are all isomorphic to Rn? 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?

Answer

For any integer k, the set Mk of complex-differentiable functions f defined on the upper-half plane {x+iy:y>0} that satisfy the equations f(z+1)=f(z),f(1/z)=zkf(z) and have limit lim is a vector space over \mathbb{C}.

Two specific elements of M_k include the functions E_4(z) = 1 + 240 \sum_{n=1}^{\infty} \sigma_3(n) e^{2\pi i n z} \in M_4 and E_8(z) = 1 + 480 \sum_{n=1}^{\infty} \sigma_7(n) e^{2\pi i nz} \in M_8. Here, \sigma_k(n) is the divisor sum \sum_{d | n} d^k.

Assuming that E_4 \in M_4 it is rather easy to show that E_4^2 \in M_8.

It can be proved that M_8 is one-dimensional, so E_4^2 is a multiple of E_8. Comparing constant coefficients tells you that they must be equal, and comparing the others gives you the formula \sigma_7(n) = \sigma_3(n) + 120 \sum_{m=1}^{n-1} \sigma_3(m) \sigma_3(n-m).

For example \sigma_7(2) = 1 + 2^7 = 1 + 2^3 + 120 and \sigma_7(3) = 1 + 3^7 = 1 + 3^3 + 120(1+2^3 + 1 + 2^3).

A lot of vector spaces like this show up in number theory. They are typically finite-dimensional but working out a basis is pretty hard (certainly harder than showing that they are finite-dimensional).

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