TITLE: What are some examples of infinite dimensional vector spaces?
QUESTION [53 upvotes]: I would like to have some examples of infinite dimensional vector spaces that help me to break my habit of thinking of $\mathbb{R}^n$ when thinking about vector spaces.

REPLY [3 votes]: I'm surprised to see that no one mentioned $\mathbb{R}$ over the field $\mathbb{Q}$ as a vector  space is infinite dimensional. Here is the proof.