Let $A$ be a unital Banach algebra with the unit $1$. It is well-known that the elements in $G(A)=\{x\in A: \|1-x\|<1\}$ are all invertible, called the general linear group.
Such a property does not hold in unital normed algebras in general. For example consider the space of all polynomials on the unit interval.
Q. Any example of a non-complete unital normed algebra $A$ for which $G(A)$ is contained in $A^{-1}$?
Take $A$ to be continuously differentiable functions on $[0,1]$ provided with the supremum norm. If $\|1-x\| <1$ then $x$ is invertible, and $x^{-1}$ is continuously differentiable by standard calculus rules, that is, $x^{-1} \in A$.