Differentiable function in functional analysis
In the mathematical discipline of functional analysis, a differentiable vector-valued function from Euclidean space is a differentiable function valued in a topological vector space (TVS) whose domains is a subset of some finite-dimensional Euclidean space. It is possible to generalize the notion of derivative to functions whose domain and codomain are subsets of arbitrary topological vector spaces (TVSs) in multiple ways. But when the domain of a TVS-valued function is a subset of a finite-dimensional Euclidean space then many of these notions become logically equivalent resulting in a much more limited number of generalizations of the derivative and additionally, differentiability is also more well-behaved compared to the general case. This article presents the theory of -times continuously differentiable functions on an open subset of Euclidean space (), which is an important special case of differentiation between arbitrary TVSs. This importance stems partially from the fact that every finite-dimensional vector subspace of a Hausdorff topological vector space is TVS isomorphic to Euclidean space so that, for example, this special case can be applied to any function whose domain is an arbitrary Hausdorff TVS by restricting it to finite-dimensional vector subspaces.
All vector spaces will be assumed to be over the field where is either the real numbers or the complex numbers
Continuously differentiable vector-valued functions
A map which may also be denoted by between two topological spaces is said to be -times continuously differentiable or if it is continuous. A topological embedding may also be called a -embedding.
Curves
Differentiable curves are an important special case of differentiable vector-valued (i.e. TVS-valued) functions which, in particular, are used in the definition of the Gateaux derivative. They are fundamental to the analysis of maps between two arbitrary topological vector spaces and so also to the analysis of TVS-valued maps from Euclidean spaces, which is the focus of this article.
A continuous map from a subset that is valued in a topological vector space is said to be (once or -time) differentiable if for all it is differentiable at which by definition means the following limit in exists:
where in order for this limit to even be well-defined,
must be an accumulation point of
If
is differentiable then it is said to be
continuously differentiable or
if its
derivative, which is the induced map
is continuous. Using induction on
the map
is
-times continuously differentiable or
if its
derivative
is continuously differentiable, in which case the
-derivative of is the map
It is called
smooth,
or
infinitely differentiable if it is
-times continuously differentiable for every integer
For
it is called
-times differentiable if it is
-times continuous differentiable and
is differentiable.
A continuous function from a non-empty and non-degenerate interval into a topological space is called a curve or a curve in A path in is a curve in whose domain is compact while an arc or C0-arc in is a path in that is also a topological embedding. For any a curve valued in a topological vector space is called a -embedding if it is a topological embedding and a curve such that for every where it is called a -arc if it is also a path (or equivalently, also a -arc) in addition to being a -embedding.
Differentiability on Euclidean space
The definition given above for curves are now extended from functions valued defined on subsets of to functions defined on open subsets of finite-dimensional Euclidean spaces.
Throughout, let be an open subset of where is an integer. Suppose and is a function such that with an accumulation point of Then is differentiable at if there exist vectors in called the partial derivatives of at , such that
where
If
is differentiable at a point then it is continuous at that point. If
is differentiable at every point in some subset
of its domain then
is said to be (
once or
-time)
differentiable in , where if the subset
is not mentioned then this means that it is differentiable at every point in its domain. If
is differentiable and if each of its partial derivatives is a continuous function then
is said to be (
once or
-time)
continuously differentiable or
For
having defined what it means for a function
to be
(or
times continuously differentiable), say that
is
times continuously differentiable or that
is if
is continuously differentiable and each of its partial derivatives is
Say that
is
smooth,
or
infinitely differentiable if
is
for all
The
support of a function
is the closure (taken in its domain
) of the set
Spaces of Ck vector-valued functions
In this section, the space of smooth test functions and its canonical LF-topology are generalized to functions valued in general complete Hausdorff locally convex topological vector spaces (TVSs). After this task is completed, it is revealed that the topological vector space that was constructed could (up to TVS-isomorphism) have instead been defined simply as the completed injective tensor product of the usual space of smooth test functions with
Throughout, let be a Hausdorff topological vector space (TVS), let and let be either:
- an open subset of where is an integer, or else
- a locally compact topological space, in which case can only be
Space of Ck functions
For any let denote the vector space of all -valued maps defined on and let denote the vector subspace of consisting of all maps in that have compact support. Let denote and denote Give the topology of uniform convergence of the functions together with their derivatives of order on the compact subsets of Suppose is a sequence of relatively compact open subsets of whose union is and that satisfy for all Suppose that is a basis of neighborhoods of the origin in Then for any integer the sets:
form a basis of neighborhoods of the origin for
as
and
vary in all possible ways. If
is a countable union of compact subsets and
is a
Fréchet space, then so is
Note that
is convex whenever
is convex. If
is
metrizable (resp.
complete,
locally convex,
Hausdorff) then so is
If
is a basis of continuous seminorms for
then a basis of continuous seminorms on
is:
as
and
vary in all possible ways.
Space of Ck functions with support in a compact subset
The definition of the topology of the space of test functions is now duplicated and generalized. For any compact subset denote the set of all in whose support lies in (in particular, if then the domain of is rather than ) and give it the subspace topology induced by If is a compact space and is a Banach space, then becomes a Banach space normed by Let denote For any two compact subsets the inclusion
is an embedding of TVSs and that the union of all
as
varies over the compact subsets of
is
Space of compactly support Ck functions
For any compact subset let
denote the inclusion map and endow
with the strongest topology making all
continuous, which is known as the
final topology induced by these map. The spaces
and maps
form a
direct system (directed by the compact subsets of
) whose limit in the category of TVSs is
together with the injections
The spaces
and maps
also form a
direct system (directed by the total order
) whose limit in the category of TVSs is
together with the injections
Each embedding
is an embedding of TVSs. A subset
of
is a neighborhood of the origin in
if and only if
is a neighborhood of the origin in
for every compact
This direct limit topology (i.e. the final topology) on
is known as the
canonical LF topology.
If is a Hausdorff locally convex space, is a TVS, and is a linear map, then is continuous if and only if for all compact the restriction of to is continuous. The statement remains true if "all compact " is replaced with "all ".
Properties
Identification as a tensor product
Suppose henceforth that is Hausdorff. Given a function and a vector let denote the map defined by This defines a bilinear map into the space of functions whose image is contained in a finite-dimensional vector subspace of this bilinear map turns this subspace into a tensor product of and which we will denote by Furthermore, if denotes the vector subspace of consisting of all functions with compact support, then is a tensor product of and
If is locally compact then is dense in while if is an open subset of then is dense in
See also
Notes
Citations
References
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