function space

Here is the example that makes abstraction worth the trouble: the functions themselves are the vectors. Fix a set X and a field F, and let V be all functions f: X -> F. Add two functions pointwise, (f + g)(x) = f(x) + g(x), and scale by (a*f)(x) = a*f(x). Check the eight axioms — they hold automatically because F already obeys them at every point — so V is a vector space whose 'arrows' are entire functions.

Restricting which functions you allow gives a parade of familiar players. Polynomials of degree at most n form an (n+1)-dimensional space with basis 1, x, x^2, ..., x^n. ALL polynomials form a space with the infinite basis 1, x, x^2, ... and no finite one. Continuous functions on [0,1], differentiable functions, solutions of a fixed linear differential equation — each is a function space, and each inherits every theorem proved from the axioms.

This is the canonical infinite-dimensional example, and it is where linear algebra stops being about grids of numbers and starts powering analysis. Differentiation D and integration are linear maps between function spaces; an eigenfunction of D is a function f with Df = lambda*f, namely e^(lambda*x). Fourier series expand a function in an orthonormal 'basis' of sines and cosines. The language of Vol I, applied to functions, becomes the engine of modern mathematics and physics.

One honest caveat: in infinite dimensions the comfortable facts can break. The polynomials have a basis you can write down, but the space of all continuous functions does not have any basis you could ever exhibit — only an abstract one guaranteed by Zorn's lemma (a Hamel basis). Function spaces are where you first feel the ground shift between finite and infinite dimension.