Exceptions in infinite-dimensional spaces

Question

What are the properties that are true in finite-dimensional spaces but fails in the infinite-dimensional space?

For example, the closed unit ball is compact only in finite-dimensional normed space.

Answer 1

There are many theorems that follow the following format:

Let $X$ be an object with dimension $n$ over minimum instance $X_0$. Then $X$ is equivalent to [canonical example] and in particular, $X$ is uniquely defined by $n$ and $X_0$.

The most famous examples are vector spaces of a field (uniquely determined by base field and dimension) and field extensions of $\mathbb{F}_p$ (uniquely determined by $p$ and the degree of the extension), but there are actually tons and tons of examples of theorems with this general structure.

The vast majority of these statements are provable via the same proof schema from Model Theory. It is also provable via model theory that for most such problems, the result is false in infinite dimension.