Bloch space
Encyclopedia
In the mathematical
field of complex analysis
, the Bloch space, named after André Bloch
and denoted , is the space of holomorphic function
s f defined on the open
unit disc D in the complex plane, such that the function
is bounded. It is a Banach space
, with the norm defined by
(Bloch norm). The elements of are called Bloch functions.
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
field of complex analysis
Complex analysis
Complex analysis, traditionally known as the theory of functions of a complex variable, is the branch of mathematical analysis that investigates functions of complex numbers. It is useful in many branches of mathematics, including number theory and applied mathematics; as well as in physics,...
, the Bloch space, named after André Bloch
André Bloch (mathematician)
André Bloch was a Jewish-French mathematician who is best remembered for a result called Bloch's theorem. Bloch was institutionalized in a mental asylum for thirty-one years of his life, during which all of his mathematical output was produced.- Early life :Bloch was born in 1893 in Besançon,...
and denoted , is the space of holomorphic function
Holomorphic function
In mathematics, holomorphic functions are the central objects of study in complex analysis. A holomorphic function is a complex-valued function of one or more complex variables that is complex differentiable in a neighborhood of every point in its domain...
s f defined on the open
Open set
The concept of an open set is fundamental to many areas of mathematics, especially point-set topology and metric topology. Intuitively speaking, a set U is open if any point x in U can be "moved" a small amount in any direction and still be in the set U...
unit disc D in the complex plane, such that the function
is bounded. It is a Banach space
Banach space
In mathematics, Banach spaces is the name for complete normed vector spaces, one of the central objects of study in functional analysis. A complete normed vector space is a vector space V with a norm ||·|| such that every Cauchy sequence in V has a limit in V In mathematics, Banach spaces is the...
, with the norm defined by
(Bloch norm). The elements of are called Bloch functions.