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We take a complementary view to the Auslander-Reiten trend of thought: Instead of specializing a module category to the level where the existence of an almost split sequence is inferred, we explore the structural consequences that result if we assume the existence of a single almost split sequence under the most general conditions. We characterize the structure of the bimodule ${}_{\Delta }Ext{}_R{(C,A)}_{\Gamma }$ with an underlying ring $R$ solely assuming that there exists an almost split sequence of left $R$-modules $0\to A\to B\to C\to 0$. $\Delta$ and $\Gamma$ are...

Let ${\left(E_i\right)}_{i\in I}$ be a family of normed spaces and $\lambda$ a space of scalar generalized sequences. The $\lambda$-sum of the family ${\left(E_i\right)}_{i\in I}$ of spaces is $$\lambda \left\{{\left(E_i\right)}_{i\in I}\right\}:=\{{\left(x_i\right)}_{i\in I},x_i\in E_i,\text{and}(\parallel x_i{\parallel )}_{i\in I}\in \lambda \}.$$ Starting from the topology on $\lambda$ and the norm topology on each $E_i,$ a natural topology on $\lambda \left\{{\left(E_i\right)}_{i\in I}\right\}$ can be defined. We give conditions for $\lambda \left\{{\left(E_i\right)}_{i\in I}\right\}$ to be quasi-barrelled, barrelled or locally complete.