I've just started a course in Representation Theory, and in solving our first homework I've used a couple of theorems about finite-dimensional vector spaces (for an example, rank-nullity theorem). My colleagues pointed out to me that we are working in general vector spaces, so I've patched up places where I've used those theorems with more general arguments.
So, why did I make that mistake? Well, in my previous linear algebra courses we mostly worked with finite-dimensional vector spaces, so in my mind I started to consider all vector spaces finite-dimensional.
To fix that, and to prevent future mishaps, I would like to see a few differences between finite-dim. and infinite-dim. vector spaces. More 'obvious' fact is in finite-dimensional space, the better.