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.