Norm of Inverse of Sequence of Invertible Elements Converging to Non-Invertible Element in Unital Banach Algebra

Theorem

Let $\struct {A, \norm {\, \cdot \,} }$ be a unital Banach algebra.

Let $\map G A$ be the group of units of $A$.

Let $x \in A \setminus \map G A$.

Let $\sequence {x_n}_{n \in \N}$ be a sequence in $\map G A$ such that $x_n \to x$.

Then $\norm {x_n^{-1} } \to \infty$ as $n \to \infty$.

Proof

Note that if there existed $n \in \N$ such that:

$\ds \norm {x - x_n} < \frac 1 {\norm {x_n^{-1} } }$

then we would have $x \in \map G A$ from Group of Units in Unital Banach Algebra is Open.

So, we have:

$\ds \frac 1 {\norm {x_n^{-1} } } \le \norm {x - x_n}$ for each $n \in \N$.

Since we have $x_n \to x$, we have $\norm {x - x_n} \to 0$, and hence:

$\ds \frac 1 {\norm {x_n^{-1} } } \to 0$ as $n \to \infty$.

So:

$\norm {x_n^{-1} } \to \infty$ as $n \to \infty$.

$\blacksquare$