What are some examples of infinite dimensional vector spaces?

Question

I would like to have some examples of infinite dimensional vector spaces that help me to break my habit of thinking of $R^n$ when thinking about vector spaces.

Answer 1

My go-to is thinking of polynomials. In this case, your basis vectors are the functions {1, x, x², x³, .....}. Any polynomial you can think of can be written in terms of this basis:

4x⁵-3x³+5 = [5, 0, 0, -3, 0, 4, 0, 0, 0, ....].

Incidentally this allows you to conceptualize derivatives as a linear transformation. If you take any polynomial written as a (column) vector, you can multiply it by the matrix below to get its derivative vector:

[0, 1, 0, 0, 0, ....]

[0, 0, 2, 0, 0, ....]

[0, 0, 0, 3, 0, ....] x = x'

[0, 0, 0, 0, 4, ....]

[........................]

The nonzero numbers are the exponents that come down from different powers of x, and the fact that they lie off the diagonal means that you are bumped down to the next power of x in line. The first column of zeros is what multiplies the constant part of the polynomial and makes it vanish.

If you've covered them in differential equations, the Fourier series of a function is also an infinite-dimensional vector space. This time the basis functions are {1, sin(x), cos(x), sin(2x), cos(2x), sin(3x), cos(3x), ...}. The Fourier coefficients would constitute the values used in a vector.