Differences between infinite-dimensional and finite-dimensional vector spaces

(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.

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

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