Characterizations of almost shrinking bases

Abstract

The material of this paper depends upon the theory of locally convex spaces, sequence spaces and Schauder bases in topological vector spaces and as such we refer to [2] (cf. also [9]), [6] and [7] respectively for several unexpalined definitions, results and terms prevalent in the sequel.However, we do recall a few definitions and terms relevant to the present paper. So, we write thorughout $X ≡(X, T)$ for an arbitrary Hausdorff locally convex space (l.c.s.) with $X^*$ denoting the topological dual of X and $D_T$ representing the saturated collection of all T-continuous seminorms generating the locally convex (l.c) topology T on X.Also we write the pair of sequences ${x_n; f_n}$ for an arbitrary Schauder basis (S.b.) for X where $x_n∈ X, f_n∈ X^*$ and $f_m(x_n)=δ_{mn}; m,n≥1.$ An S.b. ${x_n;f_n}$ for $(X,T)$ is called shrinking if ${f_n;\Psi x_n}$ is an S.b. for the strong dual $(X^*, ?(X^*,X)),\Psi$ being the usual canonical embedding from X into $X^**≡ (X^*,?(X^*,X))^*.$