Differentiable vector-valued functions from Euclidean space


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 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 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 ' or ' if it is continuous. A topological embedding may also be called a .

Curves

Differentiable curves are an important special case of differentiable vector-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 ' if for all it is ' 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 ' or ' if its ', which is the induced map is continuous.
Using induction on the map is ' or ' if its derivative is continuously differentiable, in which case the ' is the map
It is called ', or ' if it is -times continuously differentiable for every integer
For it is called ' 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 ' or a ' in
A ' in is a curve in whose domain is compact while an ' or ' in is a path in that is also a topological embedding.
For any a curve valued in a topological vector space is called a ' if it is a topological embedding and a curve such that for every where it is called a ' if it is also a path 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 ' if there exist vectors in called the ', 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 ', 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 ' or '
For having defined what it means for a function to be, say that is ' or that ' if is continuously differentiable and each of its partial derivatives is
Say that is ', or ' if is for all
The ' of a function is the closure of the set

Spaces of ''C''''k'' 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. After this task is completed, it is revealed that the topological vector space that was constructed could 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, let and let be either:
  1. an open subset of where is an integer, or else
  2. a locally compact topological space, in which case can only be

Space of ''C''''k'' 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 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 ''C''''k'' 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 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 ''C''''k'' 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 whose limit in the category of TVSs is together with the injections
The spaces and maps also form a direct system 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 on is known as the .
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 ".

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