A function
acting from some domain
of a Banach space
into a Banach space
that is
differentiable according to Fréchet
everywhere in
,
i.e. is such that for any point
there exists a bounded linear operator
from
into
for which the following relation is true:
where
denotes the norm on
or on
;
is called the
Fréchet differential
of
at
.
Another approach to the notion of an abstract analytic function
is based on differentiability according to Gâteaux. A function
from
into
is
weakly analytic
in
,
or
differentiable according to Gâteaux
in
,
if for each continuous linear functional
on
and each element
the complex function
is a holomorphic function of the complex variable
in the disc
,
where
.
Any abstract analytic function in a domain
is continuous and weakly analytic in
.
The converse proposition is also true, and the continuity condition can
be replaced by local boundedness or by continuity according to Baire.
The term
"abstract analytic function"
is sometimes employed in
a narrower sense, when it means a function
of a complex variable
with values in a Banach space or even in a locally convex linear topological space
.
In such a case any weakly analytic function
in a domain
of the complex plane
is an abstract analytic function. One can also say that a function
is an abstract analytic function in a domain
if and only if
is continuous in
and if for any simple closed rectifiable contour
the integral
vanishes. For an abstract analytic function
of a complex variable
the
Cauchy formula
(cf.
Cauchy integral)
is valid.
Let
be a weakly analytic function in a domain
of a Banach space
.
Then
,
as a function of the complex variable
,
has derivatives of all orders in the domain
,
,
these derivatives being abstract analytic functions from
into
.
If the set
belongs to
,
then
where the series converges in norm, and
A function
from
into
is called a
polynomial
with respect to the variable
of degree at most
if, for all
and for all complex
,
one has
where the functions
are independent of
.
The degree of
is exactly
if
.
A
power series
is a series of the form
where
are homogeneous polynomials of degree
so that
,
,
for all complex
.
An arbitrary weakly convergent power series
in a domain
converges in norm towards some weakly analytic function
in
,
and
,
.
A function
is an abstract analytic function if and only if it can be
developed in a power series in a neighbourhood of all points
where all
are continuous in
.
Many fundamental results in the classical theory of analytic functions
— such as the
maximum-modulus principle,
the uniqueness theorems, the
Vitali theorem,
the
Liouville theorem,
etc. — are applicable to abstract analytic functions
if suitable changes are introduced. The set of all analytic functions in a domain
forms a linear space.
The notion of an abstract analytic function can be generalized to wider classes of spaces
and
,
such as locally convex topological spaces, Banach spaces
over an arbitrary complete valuation field, etc.