Sufficient conditions are given in order that, for a bounded closed convex subset $B$ of a locally convex space $E$, the set $C(X,B)$ of continuous functions from the compact space $X$ into $B$, is the uniformly closed convex hull in $C(X,E)$ of its extreme points. Applications are made to the unit ball of bounded (or compact, or weakly compact) operators from certain Banach spaces into $C\left(X\right)$.

Standard facts about separating linear functionals will be used to determine how two cones $C$ and $D$ and their duals $C^{*}$ and $D^{*}$ may overlap. When $TV\to W$ is linear and $K\subset V$ and $D\subset W$ are cones, these results will be applied to $C=T\left(K\right)$ and $D$, giving a unified treatment of several theorems of the alternate which explain when $C$ contains an interior point of $D$. The case when $V=W$ is the space $H$ of $n\times n$ Hermitian matrices, $D$ is the $n\times n$ positive semidefinite matrices, and $T\left(X\right)=AX+X^{*}A$ yields new and known results about the existence of block diagonal...

We examine the so-called three-space-stability for some classes of linear topological and locally convex spaces for which this problem has not been investigated.

The aim of this paper is an investigation of topological algebras with an orthogonal sequence which is total. Closed prime ideals or closed maximal ideals are kernels of multiplicative functionals and the continuous multiplicative functionals are given by the “coefficient functionals”. Our main result states that an orthogonal total sequence in a unital Fréchet algebra is already a Schauder basis. Further we consider algebras with a total sequence ${\left(x_n\right)}_{n\in \mathbb{N}}$ satisfying $x_n^2=x_n$ and $x_n{x}_{n+1}=x_{n+1}$ for all n ∈ ℕ.

It is shown that all maximal regular ideals in a Hausdorff topological algebra A are closed if the von Neumann bornology of A has a pseudo-basis which consists of idempotent and completant absolutely pseudoconvex sets. Moreover, all ideals in a unital commutative sequentially Mackey complete Hausdorff topological algebra A with jointly continuous multiplication and bounded elements are closed if the von Neumann bornology of A is idempotently pseudoconvex.

We prove that each non-separable completely metrizable convex subset of a Fréchet space is homeomorphic to a Hilbert space. This resolves a more than 30 years old problem of infinite-dimensional topology. Combined with the topological classification of separable convex sets due to Klee, Dobrowolski and Toruńczyk, this result implies that each closed convex subset of a Fréchet space is homeomorphic to $[0,1]ⁿ\times {[0,1)}^m\times \ell ₂\left(\kappa \right)$ for some cardinals 0 ≤ n ≤ ω, 0 ≤ m ≤ 1 and κ ≥ 0.

We show that the strong dual X’ to an infinite-dimensional nuclear (LF)-space is homeomorphic to one of the spaces: ${ℝ}^{\omega }$, ${ℝ}^{\infty }$, $Q\times {ℝ}^{\infty }$, ${ℝ}^{\omega }\times {ℝ}^{\infty }$, or ${\left({ℝ}^{\infty }\right)}^{\omega }$, where ${ℝ}^{\infty }=lim{ℝ}^n$ and $Q={[-1,1]}^{\omega }$. In particular, the Schwartz space D’ of distributions is homeomorphic to ${\left({ℝ}^{\infty }\right)}^{\omega }$. As a by-product of the proof we deduce that each infinite-dimensional locally convex space which is a direct limit of metrizable compacta is homeomorphic either to ${ℝ}^{\infty }$ or to $Q\times {ℝ}^{\infty }$. In particular, the strong dual to any metrizable infinite-dimensional Montel space is homeomorphic either...

Let $L^{\varphi }\left(X\right)$ be an Orlicz-Bochner space defined by an Orlicz function $\varphi$ taking only finite values (not necessarily convex) over a $\sigma$-finite atomless measure space. It is proved that the topological dual $L^{\varphi }{\left(X\right)}^{*}$ of $L^{\varphi }\left(X\right)$ can be represented in the form: $L^{\varphi }{\left(X\right)}^{*}=L^{\varphi }{\left(X\right)}_n^{\sim }\oplus L^{\varphi }{\left(X\right)}_s^{\sim }$, where $L^{\varphi }{\left(X\right)}_n^{\sim }$ and $L^{\varphi }{\left(X\right)}_s^{\sim }$ denote the order continuous dual and the singular dual of $L^{\varphi }\left(X\right)$ respectively. The spaces $L^{\varphi }{\left(X\right)}^{*}$, $L^{\varphi }{\left(X\right)}_n^{\sim }$ and $L^{\varphi }{\left(X\right)}_s^{\sim }$ are examined by means of the H. Nakano’s theory of conjugate modulars. (Studia Mathematica 31 (1968), 439–449). The well known results of the duality theory...

In this article, we formalize topological properties of real normed spaces. In the first part, open and closed, density, separability and sequence and its convergence are discussed. Then we argue properties of real normed subspace. Then we discuss linear functions between real normed speces. Several kinds of subspaces induced by linear functions such as kernel, image and inverse image are considered here. The fact that Lipschitz continuity operators preserve convergence of sequences is also refered...