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.
-times continuously differentiable functions on an open subset
), 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.
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.
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.
is not mentioned then this means that it is differentiable at every point in its domain.
is differentiable and if each of its partial derivatives is a continuous function then
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
be a Hausdorff topological vector space (TVS), let
the topology of uniform convergence of the functions together with their derivatives of order
complete, locally convex, Hausdorff) then so is
[1] The definition of the topology of the space of test functions is now duplicated and generalized.
) and give it the subspace topology induced by
denote the inclusion map and endow
continuous, which is known as the final topology induced by these map.
be a Hausdorff locally convex topological vector space and for every continuous linear form
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
is a complete Hausdorff locally convex space, then
is canonically isomorphic to the injective tensor product