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The finite-dimensional decomposition property in non-Archimedean Banach spaces. (English) Zbl 1315.46082

Let \((K,|\cdot|)\) be a non-Archimedean (NA)-valued field and \((E,\|\cdot\|)\) a NA Banach space over \(K\). This means that \(|\alpha+\beta|\leq\max\{|\alpha|,|\beta|\}\), \(\alpha,\beta\in K\), and \(\|x+y\|\leq\max\{\|x\|,\|y\|\}\), \(x,y\in E\), i.e., the so-called ultrametric inequality is satisfied by the absolute value and by the norm. For a subset \(X\) of \(E\), one denotes by \([X]\) the linear subspace of \(E\) generated by \(X\). The space \(E\) is called of countable type if there exists a countable set \(X\subset E\) such that \([X]\) is dense in \(E\). In this case, every closed subspace of \(E\) is also of countable type. Two subspaces \(D_1\), \(D_2\) of \(E\) are called orthogonal, \(D_1\perp D_2\), if \(\|d_1+d_2\|=\max\{\|d_1\|,\|d_2\|\}\) for all \((d_1,d_2)\in D_1\times D_2\). For \(x,y\neq 0\), the notation \(x\perp y\) means that \([x]\perp[y]\). One says that the space \(E\) has the orthogonal finite-dimensional decomposition property (OFDDP) if there exists a sequence \((D_n)\) of finite-dimensional subspaces of \(E\) with \(D_n\perp D_m\) for \(n\neq m\), such that every \(x\in E\) can be uniquely written as \(x=\sum_nx_n\) with \(x_n\in D_n\), \(n\in\mathbb N\). A sequence \((x_n)\) in \(E\) is called an orthogonal basis of \(E\) if \(x_n\perp x_m\) for all \(n\neq m\) and every \(x\in E\) can be uniquely written as \(x=\sum_n\alpha_nx_n.\) There are several books on NA functional analysis, as, for instance, the recent one by C. Perez-Garcia and W. H. Schikhof [Locally convex spaces over non-Archimedean valued fields. Cambridge: Cambridge University Press (2010; Zbl 1193.46001)], which can be recommended as an introduction to the area and for recent results as well.
In the NA case, every Banach space of countable type has a Schauder basis, even an orthogonal basis, if the field \(K\) is spherically complete. If \(K\) is not spherically complete, then there exist finite-dimensional NA Banach spaces over \(K\) having no orthogonal bases [A. Kubzdela, Contemp. Math. 384, 169–185 (2005; Zbl 1094.46047)].
The main result of this paper asserts that there exists an NA Banach space of countable type having the OFDDP and containing a closed subspace of codimension one without the OFDDP, thus answering in the negative a question raised by C. Perez-Garcia and W. H. Schikhof [Glas. Mat., III. Ser. 49, No. 2, 407–419 (2014; Zbl 1320.46059)]. The paper also contains some conditions under which finite-codimensional subspaces of an NA Banach space of countable type with OFDDP also have the OFDDP.

MSC:

46S10 Functional analysis over fields other than \(\mathbb{R}\) or \(\mathbb{C}\) or the quaternions; non-Archimedean functional analysis
46B15 Summability and bases; functional analytic aspects of frames in Banach and Hilbert spaces
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