It is proved that there is a unique metrizable simplex $S$ whose extreme points are dense. This simplex is homogeneous in the sense that for every 2 affinely homeomorphic faces $F_1$ and $F_2$ there is an automorphism of $S$ which maps $F_1$ onto $F_2$. Every metrizable simplex is affinely homeomorphic to a face of $S$. The set of extreme points of $S$ is homeomorphic to the Hilbert space ${\ell }_2$. The matrices which represent $A\left(S\right)$ are characterized.

An explicit representation of the n-fold symmetric tensor product (equipped with a natural topology τ such as the projective, injective or inductive one) of the finite direct sum of locally convex spaces is presented. The formula for ${⨂}_{\tau ,s}^n\left(F_1⨁F_2\right)$ gives a direct proof of a recent result of Díaz and Dineen (and generalizes it to other topologies τ) that the n-fold projective symmetric and the n-fold projective “full” tensor product of a locally convex space E are isomorphic if E is isomorphic to its square $E^2$.

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.

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 be an operator ideal on LCS’s. A continuous seminorm p of a LCS X is said to be - continuous if $Q{̃}_p{\in }^{inj}(X,X{̃}_p)$, where $X{̃}_p$ is the completion of the normed space $X_p=X/p^{-1}\left(0\right)$ and $Q{̃}_p$ is the canonical map. p is said to be a Groth()- seminorm if there is a continuous seminorm q of X such that p ≤ q and the canonical map $Q{̃}_{pq}:X{̃}_q\to X{̃}_p$ belongs to $(X{̃}_q,X{̃}_p)$. It is well known that when is the ideal of absolutely summing (resp. precompact, weakly compact) operators, a LCS X is a nuclear (resp. Schwartz, infra-Schwartz) space if and only if every continuous...