Differences between infinite-dimensional and finite-dimensional vector spaces

Question

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.

Answer 1

(1). There are endomorphisms $T$ with $\ker(T)=\{0\}$ which are not surjective.

(2). Not in every case a linear form $\phi$ is representable by a vector $v$ in presence of a scalar product, i.e., there doesn't exist a vector $v$ that $\phi(.)=\langle v,.\rangle$.

(3). Not all linear mappings are continuos.

(4). You can equip a vector space with two different norms such that the unit ball in respect to the first norm in unbounded in respect to the second.

It's just a brand new world.

Answer 2

A finite dimensional vector space is always isomorphic to its dual, but this is false for an infinite dimensional vector space.

Answer 3

For example, not every (infinite) matrix corresponds to a linear operator.