### A Characterization of Totally Reflexive Fréchet Spaces.

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It is shown in the note that every reflexive Orlicz function space has the Schroeder-Bernstein Property and the Primary Property.

In this article, we considered bidual spaces and reflexivity of real normed spaces. At first we proved some corollaries applying Hahn-Banach theorem and showed related theorems. In the second section, we proved the norm of dual spaces and defined the natural mapping, from real normed spaces to bidual spaces. We also proved some properties of this mapping. Next, we defined real normed space of R, real number spaces as real normed spaces and proved related theorems. We can regard linear functionals...

Let E and F be two vector spaces in separating duality. Let us consider T0, the uniform convergence topology on E on the partial sums of families of F which are weakly summable to 0 in F; then, if (E',T'0) is the completion of (E,T0), the finest locally convex topology T on F for which all the weakly summable families in F are also T-summable, is the uniform convergence topology on the T'0-compact subsets of E'. If F is a Banach space and E its dual space F', every weakly summable family in F is...

Fréchet spaces of strongly, weakly and weak*-continuous Fréchet space valued functions are considered. Complete solutions are given to the problems of their injectivity or embeddability as complemented subspaces in dual Fréchet spaces.

Let Λ_R(α) be a nuclear power series space of finite or infinite type with lim_{j→∞} (1/j) log α_j = 0. We consider open polydiscs D_a in Λ_R(α)'_b with finite radii and the spaces H(D_a) of all holomorphic functions on D_a under the compact-open topology. We characterize all isomorphy classes of the spaces {H(D_a) | a ∈ Λ_R(α), a > 0}. In the case of a nuclear power series space Λ₁(α) of finite type we give this characterization in terms of the invariants (Ω̅ ) and (Ω̃ ) known from the theory...

The aim of this work is to generalize lacunary statistical convergence to weak lacunary statistical convergence and $\mathcal{I}$-convergence to weak $\mathcal{I}$-convergence. We start by defining weak lacunary statistically convergent and weak lacunary Cauchy sequence. We find a connection between weak lacunary statistical convergence and weak statistical convergence.

For U open in a locally convex space E it is shown in [31] that there is a complete locally convex space G(U) such that $G{\left(U\right)}_{i}^{\text{'}}=(\mathscr{H}\left(U\right),{\tau}_{\delta})$. Here, we assume U is balanced open in a Fréchet space and give necessary and sufficient conditions for G(U) to be Montel and reflexive. These results give an insight into the relationship between the ${\tau}_{0}$ and ${\tau}_{\omega}$ topologies on ℋ (U).