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The space of real-analytic functions has no basis. (English) Zbl 0990.46015

Let \(\Omega\) be an open connected subset of the Euclidean space \(\mathbb{R}^d\). Let \(A(\Omega)\) be the space of real-analytic functions on \(\Omega\) with its usual topology. A quite interesting result is proved in this paper asserting that all metrizable complemented subspaces of \(A(\Omega)\) are finite-dimensional. A PLN-space is the projective limit of a sequence of LN-space, i.e., LB-spaces with nuclear linking maps. The space \(A(\Omega)\) is a PLN-space.
In this paper, Schauder bases for PLN-spaces are studied and, among other results, the following one is obtained:
a) Every ultrabornological PLN-space with basis is either an LN-space or it contains an infinite-dimensional Fréchet subspace. Analogously, it is either a Fréchet space or contains an infinite-dimensional complemented LN-space.
A Fréchet space \(E\) with a fundamental sequence \((\|.\|_n)\) of seminorms defining the topology is said to have property \((\overline{\overline \Omega})\) if \[ \forall k\exists m\forall n,\;0\in ]0,1[ \exists\subset\forall u\in E':\|u\|^*_m\leq C\|u\|^{*0}_\ell\|u\|^{*(1-0)}_n. \] Here \(\|.\|^*\) denotes the dual norm for \(\|.\|\). \(E\) is said to have property (DN) if \[ \exists n\forall k\exists\ell,\;C> 0,\;\tau\in ]0,1[:\|x\|_k\leq C\|x\|^\tau_m\|x\|^{1-\tau}_\ell \] for every \(x\) of \(E\).
The authors give the following results:
b) Every Fréchet space \(E\) which is a quotient of \(A(\Omega)\) has property \((\overline{\overline\Omega})\).
c) If \(\Omega\) is connected then every Fréchet subspace \(E\) of \(A(\Omega)\) has property (DN).
From b) and c) follows:
d) If \(\Omega\) is connected then every complemented Fréchet subspace of \(A(\Omega)\) is finite-dimensional. As a consequence of d).
The following main result is achieved: Let \(\Omega\) be connected. If \(E\) is a complemented subspace with basis of \(A(\Omega)\), then \(E\) is an LB-space. In particular, \(A(\Omega)\) has no basis. In an added-in-proof note, the authors prove the following: For an arbitrary open set \(\Omega\), every complemented subspace of \(A(\Omega)\) with basis is a product of LB-spaces, in particular, \(A(\Omega)\) has no basis.

MSC:

46E10 Topological linear spaces of continuous, differentiable or analytic functions
46A35 Summability and bases in topological vector spaces
46A04 Locally convex Fréchet spaces and (DF)-spaces
46A13 Spaces defined by inductive or projective limits (LB, LF, etc.)
46A45 Sequence spaces (including Köthe sequence spaces)
26E05 Real-analytic functions
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