In mathematics, especially functional analysis, a hypercyclic operator on a topological vector space X is a continuous linear operator T: X → X such that there is a vector x ∈ X for which the sequence {Tn x: n = 0, 1, 2, …} is dense in the whole space X.
In other words, the smallest closed invariant subset containing x is the whole space.
Universality in general involves a set of mappings from one topological space to another (instead of a sequence of powers of a single operator mapping from X to X), but has a similar meaning to hypercyclicity.
Examples of universal objects were discovered already in 1914 by Julius Pál, in 1935 by Józef Marcinkiewicz, or MacLane in 1952.
However, it was not until the 1980s when hypercyclic operators started to be more intensively studied.