All Questions

Can there be a function $m(a,b)$ that is both associative and a mean, i.e., $\min (a,b) \leq m(a,b) \leq \max (a,b)$? The obvious solutions are $m(a,b) = \max(a,b)$ or $\min(a,b)$, but are there ...

Given a real vector space $V$ which is the union of a totally ordered family of vector subspaces $V=\bigcup_{i\in I} V_i$. By that I mean that we assume that $(I,\leq)$ is a totally ordered set and ...