Abstract: A little 1960s book Some Random Series of Functions by Jean-Pierre Kahane ignited a new mathematical discipline in the intersection of probability and functional analysis - probability theory on Banach spaces. Of course, the factual origins can be traced many decades back.

A classical problem: a Fourier series is transformed by inserting arbitrary signs at the coefficients. Does this Hilbertian isometry extend to Lp-spaces for p distinct from 2? Can the signs be arbitrary, or are they subject to some constraints? Yes, they are: almost all but not all signs are admissible. This is one of many ways random series in a Banach space arise. Random signs can be further extended to random multipliers, and so probability on Banach spaces is born.

In the talk I will address a quite interesting category of random spaces where the classical rules are reversed. For example, Jensen's or Chebyshev's inequalities will hold in reverse, entailing equivalences of vector topologies that normally are strictly distinct.