Non unicité des topologies localement-convexes complètes sur certaines algèbres

We are interested, here, by the problems 1, 8 and 9 due to Wojciechowski and Żelazko (1997). We show that every algebra A generated by ℵ elements, such that ℵ > ℵ0, can be a locally-convex (resp. complete locally p-convex, p ϵ (0, 1]) semitopological algebra in ℵ +1 (resp. max{2ℵ0, ℵ +1}) different ways, which reduce the problem 9. Moreover if ℵℵ0 = ℵ, A can be a complete locally-convex semitopological algebra in ℵ +1 different ways, which solves the problem 1 for every ℵℵ0-generated algebra. On the other hand, we prove that for every algebra of polynomials generated by ℵ indeterminates, ℵ ≥ ℵ0 we can assign max{2ℵ0, ℵ +1} distinct topologies which make it a locally-convex topological algebra. Then, we get a reduction of problem 8.