Normed representation of a Banach algebra ($\pi (A)\subseteq \mathcal{B}(X)$)

A normed representation acts by individually bounded operators on a normed vector space. This condition alone does not assume that $a\mapsto \pi (a)$ is continuous in the operator norm, nor that the representation space is complete. A continuous representation has a common bound $\Vert \pi (a)\Vert \le C\Vert a\Vert$.

Every algebraically irreducible representation of a Banach algebra that is normed is continuous. The continuous orbit vectors form an invariant subspace. If all nonzero orbit maps were discontinuous, apply the countable annihilation sequence in an irreducible Banach module and the gliding-hump continuity principle to right multiplication on the algebra and evaluation of the operator-valued representation. Off-diagonal maps vanish, whereas diagonal maps are discontinuous, a contradiction. Once every orbit map is continuous, the Uniform boundedness principle on the Banach algebra gives the common operator-norm bound. Completing the original normed representation space supplies the Banach operator space needed in the argument.