If the logarithmic dimension of a Cantor-type set K is smaller than $1$ , then the Whitney space $\mathcal {E}(K)$ possesses an interpolating Faber basis. For any generalized Cantor-type set K, a basis in $\mathcal {E}(K)$ can be presented by means of functions that are polynomials locally. This gives a plenty of bases in each space $\mathcal {E}(K)$ . We show that these bases are quasi-equivalent.

The non-archimedean power series spaces, ${{A}_{1}}(a)\,\text{and}\,{{A}_{\infty }}(b)$, are the best known and most important examples of non-archimedean nuclear Fréchet spaces. We prove that the range of every continuous linear map from ${{A}_{p}}(a)\,\text{to}\,{{A}_{q}}(b)$ has a Schauder basis if either $p\,=\,1$ or $p\,=\,\infty $ and the set ${{M}_{b,a}}$ of all bounded limit points of the double sequence (${{({{b}_{i}}/{{a}_{j}})}_{i,j\in \mathbb{N}}}$ is bounded. It follows that every complemented subspace of a power series space ${{A}_{p}}(a)$ has a Schauder basis if either $p\,=\,1$ or $p\,=\,\infty $ and the set ${{M}_{a,a}}$ is bounded.

A basis $\{x_n\}$ for a Hilbert space H is called a Riesz basis if it has the property that $\sum a_nx_n$ converges in H if and only if $\sum|a_n|^2<\infty$, and hence if and only if $\{x_n\}$ is the isomorphic image of some orthonormal basis for H. A consequence of a classical result of Bary [1] is that any basis for H that is quadratically near an orthonormal basis must be a Riesz basis. Motivated by this result, we study in this paper the class of normalized bases in a Hilbert space that are quadratically near some orthonormal basis, bases we call almost-orthonormal bases. In particular, we prove that any such basis must be quadratically near its Gram-Schmidt orthonormalization, and derive an internal characterization of these bases that indicates how restrictive the property of being almost-orthonormal is.

It is known that any non-archimedean Fréchet 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 Fréchet space $U$ with a basis $({{u}_{n}})$ such that any basis $({{x}_{n}})$ in a non-archimedean Fréchet space $X$ is equivalent to a subbasis $({{u}_{kn}})$ of $({{u}_{n}})$. Then any non-archimedean Fréchet space with a basis is isomorphic to a complemented subspace of $U$. In contrast to this, we show that a non-archimedean Fréchet 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 Fréchet space $H$ with a basis $({{h}_{n}})$ such that any basis $({{y}_{n}})$ in a nuclear non-archimedean Fréchet space $Y$ is equivalent to a subbasis $({{h}_{kn}})$ of $({{h}_{n}})$.

A Toeplitz decomposition of a locally convez space E into subspaces (Ek) with continuous projections (Pk) is a decomposition of every x ∈ E as x = ΣkPkx where ordinary summability has been replaced by summability with respect to an infinite and row-finite matrix. We extend to the setting of Toeplitz decompositions a number of results about the locally convex structure of a space with a Schauder decomposition. Namely, we give some necessary or sufficient conditions for being reflexive, a Montel space or a Schwartz space. Roughly speaking, each of these locally convex properties is linked to a property of the convergence of the decomposition. We apply these results to study some structural questions in projective tensor products and spaces with Cesàro bases.

If Y is a subspace of a Banach space X, either Y contains an isomorphic copy of ℓ1 or each weak Cauchy sequence in X/Y has a subsequence that is the image under the quotient mapping of a weak Cauchy sequence in X. If X is weakly sequentially complete and Y is reflexive, X/Y is weakly sequentially complete. Related structural results are given.