EDB — 118

[118] Topics:total convergence.Prerequisites:[0N8]↺↻,[0NC]↺↻,[0NF]↺↻.

Let \(V\) be a vector space with a norm \(\| x\| \); So \(V\) is also a metric space with the metric \(d(x,y)=\| x-y\| \). Show that the following two clauses are equivalent.

• \((V,d)\) is complete.

• For each sequence \((v_ n)_ n⊂ V \) such that \(∑_ n\| v_ n\| {\lt}∞\), the series \(∑_ n v_ n \) converges.

(The second is sometimes called the ”total convergence criterion”)

A normed vector space \((V,|\cdot \| )\) such that the associated metric space \((V,d)\) is complete, is called a Banach space.