We study the connection between intersection properties of balls and the existence of large faces of the unit ball in Banach spaces. Hanner’s result that a real space has the 3.2 intersection property if an only if disjoint faces of the unit ball are contained in parallel hyperplanes is extended to infinite dimensional spaces. It is shown that the space of compact operators from a space $X$ to a space $Y$ has the 3.2 intersection property if and only if $X$ and $Y$ have the 3.2 intersection property and...

We study conditions on an infinite dimensional separable Banach space $X$ implying that $X$ is the only non-trivial invariant subspace of $X^{**}$ under the action of the algebra $𝔸\left(X\right)$ of biconjugates of bounded operators on $X$: $𝔸\left(X\right)=\{T^{**}T\in ℬ\left(X\right)\}$. Such a space is called simple. We characterize simple spaces among spaces which contain an isomorphic copy of $c_0$, and show in particular that any space which does not contain ${\ell }_1$ and has property (u) of Pelczynski is simple.

Probabilistic normed linear spaces (briefly PNL spaces) were first studied by A. N. Serstnev in [1]. His definition was motivated by the definition of probabilistic metric spaces (PM spaces) which were introduced by K. Menger and subsequebtly developed by A. Wald, B. Schweizer, A. Sklar and others.In a previuos paper [2] we studied the relationship between two important classes of PM spaces, namely E-spaces and pseudo-metrically generated PM spaces. We showed that a PM space is pseudo-metrically...