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1. CMB 2011 (vol 55 pp. 821)
New Examples of Non-Archimedean Banach Spaces and Applications The study carried out in this paper about some new examples of
Banach spaces, consisting of certain valued fields extensions, is
a typical non-archimedean feature. We determine whether these
extensions are of countable type, have $t$-orthogonal bases, or are
reflexive.
As an application we construct, for a class of base fields, a norm
$\|\cdot\|$ on $c_0$, equivalent to the canonical supremum norm,
without non-zero vectors that are $\|\cdot\|$-orthogonal and such
that there is a multiplication on $c_0$ making $(c_0,\|\cdot\|)$
into a valued field.
Keywords:non-archimedean Banach spaces, valued field extensions, spaces of countable type, orthogonal bases Categories:46S10, 12J25 |
2. CMB 2008 (vol 51 pp. 604)
The Invariant Subspace Problem for Non-Archimedean Banach Spaces It is proved that every infinite-dimensional
non-archimedean Banach space of countable type admits a linear
continuous operator without a non-trivial closed invariant
subspace. This solves a problem stated by A.~C.~M. van Rooij and
W.~H. Schikhof in 1992.
Keywords:invariant subspaces, non-archimedean Banach spaces Categories:47S10, 46S10, 47A15 |
3. CMB 2007 (vol 50 pp. 149)
On Quotients of Non-Archimedean KÃ¶the Spaces We show that there exists a non-archimedean
Fr\'echet-Montel space $W$ with a basis and with a continuous norm
such that any non-archimedean Fr\'echet space of countable type is isomorphic
to a quotient of $W$. We also prove that any non-archimedean nuclear
Fr\'echet space is isomorphic to a quotient of some non-archimedean nuclear
Fr\'echet space with a basis and with a continuous norm.
Keywords:Non-archimedean KÃ¶the spaces, nuclear FrÃ©chet spaces, pseudo-bases Categories:46S10, 46A45 |
4. CMB 2004 (vol 47 pp. 108)
On Universal Schauder Bases in Non-Archimedean FrÃ©chet Spaces It is known that any non-archimedean Fr\'echet space of countable
type is isomorphic to a subspace of $c_0^{\mathbb{N}}$. In this
paper we prove that there exists a non-archimedean Fr\'echet space
$U$ with a basis $(u_n)$ such that any basis $(x_n)$ in a
non-archimedean Fr\'echet space $X$ is equivalent to a subbasis
$(u_{k_n})$ of $(u_n)$. Then any non-archimedean Fr\'echet space
with a basis is isomorphic to a complemented subspace of $U$. In
contrast to this, we show that a non-archimedean Fr\'echet space
$X$ with a basis $(x_n)$ is isomorphic to a complemented subspace
of $c_0^{\mathbb{N}}$ if and only if $X$ is isomorphic to one of
the following spaces: $c_0$, $c_0 \times \mathbb{K}^{\mathbb{N}}$,
$\mathbb{K}^{\mathbb{N}}$, $c_0^{\mathbb{N}}$. Finally, we prove
that there is no nuclear non-archimedean Fr\'echet space $H$ with
a basis $(h_n)$ such that any basis $(y_n)$ in a nuclear
non-archimedean Fr\'echet space $Y$ is equivalent to a subbasis
$(h_{k_n})$ of $(h_n)$.
Keywords:universal bases, complemented subspaces with bases Categories:46S10, 46A35 |