12.3 Total convergence[2CJ]
[116]Let in the following \(X\) be a normed vector space based on the real field \({\mathbb {R}}\), with norm \(\| \cdot \| \). Let \( (f_ n)_{n\in {\mathbb {N}}}\) be a sequence of elements of \(X\). The series \(∑^∞_{n=0}f_ n\) converges totally when \(∑^∞_{n=0}\| f_ n\| {\lt}∞\).
- E338
[117]Show that if the series of \((f_ n)_ n,(g_ n)_ n\) converge totally, then the series of \((f_ n+g_ n)_ n\) converges totally.
- E338
[118] Topics:total convergence.Prerequisites:9,10,11.
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.
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