Abstract. About 30 years ago a new branch of what von Neumann called “abstract analysis” was born. This event led to the solution of quite a few well known problems, that were formulated in purely classical terms. This was done after some people, studying this or that problem, realized that behind the classical norm on a given space, an essentially richer structure was hidden. This is the so-called “quantum norm”, or “abstract operator space structure”.

We begin recalling several classical problems that were clarified and solved after their reformulation in the language of quantum norms. Then we proceed to two main concepts of the area: the quantum norm and the completely bounded operator, presenting, at the same time, some examples and counterexamples. Here we shall explain, from the general aspect of what is now called quantum or non–commutative mathematics, the use of the terms “quantum norm” and “quantization of a normed space”.

In the whole quantum functional analysis one can easily distinguish three most deep and important results, that form the heart of the subject. These are the Ruan Representation Theorem (whose message is that there are no quantum spaces, save operator spaces), the Arveson/Wittstock Theorem (also called Quan- tum Hahn/Banach Theorem) and the Paulsen/Wittstock Decomposition Theorem (that reduces general completely bounded operators to two fairly concrete types). In the concluding part of the talk, we shall discuss these theorems, whose formulations are quite simple and transparent.