L0-Linear Modulus of a Random Linear Operator

We prove that there exists a uniqueL0-linear modulus for an a.s. bounded random linear operator on a specifical random normed module, which generalizes the classical case.

Extreme Version of Projectivity for Normed Modules Over Sequence Algebras

We define and study the so-called extreme version of the notion of a projective normed module. The relevant definition takes into account the exact value of the norm of the module in question, in contrast with the standard known definition that is formulated in terms of normtopology.After the discussion of the case where our normed algebra A is just C, we concentrate on the case of the next degree of complication, where A is a sequence algebra satisfying some natural conditions. The main results give a full characterization of extremely projective objects within the subcategory of the category of non-degenerate normed A-modules, consisting of the so-called homogeneous modules. We consider two cases, ‘non-complete’ and ‘complete’, and the respective answers turn out to be essentially different.In particular, all Banach non-degenerate homogeneous modules consisting of sequences are extremely projective within the category of Banach non-degenerate homogeneous modules. However, neither of them, provided it is infinite-dimensional, is extremely projective within the category of all normed non-degenerate homogeneous modules. On the other hand, submodules of these modules consisting of finite sequences are extremely projective within the latter category.

A Minimax Theorem forL-0-Valued Functions on Random Normed Modules

We generalize the well-known minimax theorems toL¯0-valued functions on random normed modules. We first give some basic properties of anL0-valued lower semicontinuous function on a random normed module under the two kinds of topologies, namely, the (ε,λ)-topology and the locallyL0-convex topology. Then, we introduce the definition of random saddle points. Conditions for anL0-valued function to have a random saddle point are given. The most greatest difference between our results and the classical minimax theorems is that we have to overcome the difficulty resulted from the lack of the condition of compactness. Finally, we, using relations between the two kinds of topologies, establish the minimax theorem ofL¯0-valued functions in the framework of random normed modules and random conjugate spaces.

THE RANDOM SUBREFLEXIVITY OF COMPLETE RANDOM NORMED MODULES

Combining respective advantages of the (ε, λ)-topology and the locally L0-convex topology we first prove that every complete random normed module is random subreflexive under the (ε, λ)-topology. Further, we prove that every complete random normed module with the countable concatenation property is also random subreflexive under the locally L0-convex topology, at the same time we also provide a counterexample which shows that it is necessary to require the random normed module to have the countable concatenation property.