Banach Algebra with Unity is Unital Banach Algebra

Theorem
Let $\mathbb F \in \set {\R, \C}$.

Let $\struct {X, \norm \cdot}$ be a non-trivial Banach algebra over $\mathbb F$.

Suppose that $X$ has an identity element $\mathbf 1_X$.

Then exists a norm $\norm \cdot '$ on $X$ such that $\struct {X, \norm \cdot '}$ is a unital Banach algebra, i.e.:
 * $\struct {X, \norm \cdot '}$ is a Banach algebra
 * $\norm {\mathbf 1_X}' = 1$