Given a complex Hilbert space 
with associated space 
of continuous linear
operators from 
to itself, the bicommutant 
of an arbitrary subset 
is the commutant of the commutant 
of 
, i.e., 
. In particular, the bicommutant is the collection
of all elements in 
that commute with all elements of 
:
 |
Across the literature on the subject, the set 
is sometimes denoted 
, a reference to the fact that a linear operator between
normed vector spaces
is continuous if and only if it is bounded (Royden and Fitzpatrick 2010). Likewise,
the bicommutant is sometimes called the double commutant.
The notions of commutant and bicommutant are fundamental to the study of von Neumann algebras (Dixmier 1981).
