There's some properties such that if satisfied by a normed space, then necessarily this normed space is finite dimensional.
An example is of course the compactness of closed bounded sets. Another example is maybe related to ordinary differential equations, it is a result of Dieudonné which states (if I am not wrong) that if the differential equation $$x'(t)=f(t,x(t))$$ has a solution for every continuous function $f:\mathbb{R}^+\times X \to X$, then $X$ must be finite dimensional.
Is there other properties like these ?
1) If a Banach space $X$ has Hamel basis with continuous coordinate functionals then it is finite dimensional.
2) If $X$ has no dense proper subspaces it is finite dimensional
3) If the weak topology and the strong topology on $X$ coincide then $X$ is finite dimensional