Are there finite dimensional $\mathbb{R}$-algebras which are not Banach algebras?

Question

It is known that not every algebra (over a ground field $\mathbb{R}$ or $\mathbb{C}$) is a Banach algebra. It might be a silly question, but are there examples of finite dimensional ($\mathbb{R}$- or $\mathbb{C}$-)algebras which are not Banach algebras (i.e. for which no submultiplicative norm exists)?

Answer 1

Every finite dimensional unitary algebra $A$ is isomorphic to a subalgebra of a matrix algebra $M_n(\mathbb R)$. Retricting the norm of the latter gives a norm on $A$.

Answer 2

I think it's clear that in the finite-dimensional case multiplication is going to be continuous with respect to any norm, just as any linear transformation is bounded. If so then any norm is going to satisfy $||xy||\le c||x||\,||y||$, and then $|||x|||=c||x||$ is submultiplicative.