Continuous norms on locally convex strict inductive limit spaces. (English) Zbl 0571.46003
Let \((E_ n)_{n\in {\mathbb{N}}}\) be a strict inductive sequence of locally convex spaces \(E_ n\) each admitting a continuous norm. The author studies the question whether the inductive limit \(E=_{n\to}E_ n\) admits a continuous norm. He proves a necessary and sufficient condition in terms of a concordance property of the continuous norms on the spaces \(E_ n\), which is satisfied e.g. if each \(E_ n\) has the countable- neighbourhood property. Furthermore he gives the following counterexample:
There exists a nuclear Fréchet space (G,\(\tau)\) with an increasing sequence of closed subspaces \(E_ n\), each having a continuous norm but the strict inductive limit \(E=_{n\to}(E_ n,\tau |_{E_ n})\) does not admit a continuous norm. It turns out that the (FN)-spaces \(E_ n\) do not have the bounded approximation property and that E does not have a Schauder basis.
There exists a nuclear Fréchet space (G,\(\tau)\) with an increasing sequence of closed subspaces \(E_ n\), each having a continuous norm but the strict inductive limit \(E=_{n\to}(E_ n,\tau |_{E_ n})\) does not admit a continuous norm. It turns out that the (FN)-spaces \(E_ n\) do not have the bounded approximation property and that E does not have a Schauder basis.
Reviewer: R.Meise
MSC:
46A13 | Spaces defined by inductive or projective limits (LB, LF, etc.) |
46A11 | Spaces determined by compactness or summability properties (nuclear spaces, Schwartz spaces, Montel spaces, etc.) |
46M40 | Inductive and projective limits in functional analysis |
46A04 | Locally convex Fréchet spaces and (DF)-spaces |
41A65 | Abstract approximation theory (approximation in normed linear spaces and other abstract spaces) |
Keywords:
continuous norm; concordance property of the continuous norms; countable- neighbourhood property; nuclear Fréchet space; strict inductive limit; (FN)-spaces; bounded approximation property; Schauder basisReferences:
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