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

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)?

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 $A$.
• @Bib-lost: You can embed $A$ in $End(A)$ since $A$ acts on itself by multiplication. Apr 3 '16 at 18:55
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.