Exercise 8, Appendix 0

Let $A$ be a complete normed algebra. Show that if $(x_n)$ is a sequence of points of $A$ such that, for every permutation $\sigma$ of $\mathbf{N}$, the sequence $(x_{\sigma(n)})$ is multipliable and its product is a unit, then each $x_n$ is a unit. (First establish the following algebraic lemma: in a ring with an identity element, if two elements $x, y$ are such that $xy$ is a unit and $yx$ is not a zero divisor, then $x$ and $y$ are both units.)

DIAGRAM OF THE PRINCIPAL TYPES OF TOPOLOGICAL SPACES

• metrizable compact sp. $\to$ locally compact sp. with countable base
• metrizable compact sp. $\to$ compact sp.
• locally compact sp. with countable base $\to$ polish sp.
• locally compact sp. with countable base $\to$ metrizable, locally compact sp.
• locally compact sp. with countable base $\to$ locally compact sp., $\sigma$-compact
• polish sp. $\to$ lusin sp.
• lusin sp. $\to$ souslin sp.
• souslin sp. $\to$ metrizable sp.
• metrizable, locally compact sp. $\to$ metrizable sp.
• metrizable, locally compact sp. $\to$ locally compact sp.
• compact sp. $\to$ locally compact sp., $\sigma$-compact
• locally compact sp., $\sigma$-compact $\to$ paracompact sp.
• locally compact sp., $\sigma$-compact $\to$ locally compact sp.
• metrizable sp. $\to$ paracompact sp.
• paracompact sp. $\to$ normal sp.
• normal sp. $\to$ completely regular sp.
• locally compact sp. $\to$ completely regular sp.
• completely regular sp. $\to$ regular sp.
• regular sp. $\to$ Hausdorff