Corollary 1

Let A be a nonzero unital normed algebra. For every $x\in A$, the spectrum Sp$_A(x)$ is nonempty.

Suppose first that A is complete. If one had Sp($x$) $=\emptyset$, the resolvent of $x$ would be holomorphic in $\mathbf{C}$ and zero at infinity, hence identically zero (VAR, R., 3.3.6, p. 29). Since $R(x, \lambda ) = (\lambda -x)^{-1}$ is invertible, it would follow that 1 = 0 and therefore that $A =\{0\}$.

In the general case, let $\widehat{A}$ be the completed algebra of A; the relation Sp$_A(x) =\emptyset$ would entail Sp$_{\widehat{A}}(x) =\emptyset$, whence $\widehat{A} =\{0\}$ and $A =\{0\}$.