Convex linear metric spaces are normable. (English) Zbl 1453.46003
The authors call a linear metric space \((X,d)\) over the field \(\mathbb R\) convex if \(d(\lambda x+(1-\lambda)y,0)\le\lambda d(x,0)+(1-\lambda)d(y,0)\) holds for all \(x,y\in X\) and each \(\lambda\), \(0\le\lambda\le1\). They prove that each (real) convex linear metric space is normable (with the norm given by \(\|x\|=d(x,0)\). The same is true in the case of complex scalars if the metric \(d\) is rotation invariant (i.e., \(d(\alpha x,0)=d(|\alpha|x,0)\) for all complex scalars \(\alpha\) and all \(x\in X\)), but not in general.
Reviewer: Zoran Kadelburg (Beograd)
MSC:
| 46A19 | Other “topological” linear spaces (convergence spaces, ranked spaces, spaces with a metric taking values in an ordered structure more general than \(\mathbb{R}\), etc.) |
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