Differentiable vector-valued functions from Euclidean space

In the field of Functional Analysis, it is possible to generalize the notion of derivative to infinite dimensional topological vector spaces (TVSs) in multiple ways. But when the domain of TVS-value functions is a subset of finite-dimensional Euclidean space then the number of generalizations of the derivative is much more limited and derivatives are more well behaved. This article presents the theory of k-times continuously differentiable functions on an open subset of Euclidean space (), which is an important special case of differentiation between arbitrary TVSs. 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

Throughout, let and let be either:

  1. an open subset of , where is an integer, or else
  2. a locally compact topological space, in which k can only be 0,

and let be a topological vector space (TVS).

Definition[1] Suppose and is a function such that with a limit point of . Then we say that f is differentiable at if there exist n vectors in Y, called the partial derivatives of f, such that
in Y
where .

Note that if f is differentiable at a point then it is continuous at that point.[1] Say that f is if it is continuous. If f is differentiable at every point in some set then we say that f is differentiable in S. If f is differentiable at every point of its domain and if each of its partial derivatives is a continuous function then we say that f is continuously differentiable or .[1] Having defined what it means for a function f to be (or k times continuously differentiable), say that f is k + 1 times continuously differentiable or that f is if f is continuously differentiable and each of its partial derivatives is . Say that f is , smooth, or infinitely differentiable if f is for all . If is any function then its support is the closure (in ) of the set .

Spaces of Ck vector-valued functions

Space of Ck functions

For any , let denote the vector space of all Y-valued maps defined on and let denote the vector subspace of consisting of all maps in that have compact support. Let denote and denote . We give the topology of uniform convergence of the functions together with their derivatives of order < k + 1 on the compact subsets of .[1] Suppose is a sequence of relatively compact open subsets of whose union is and that satisfy for all i. Suppose that is a basis of neighborhoods of the origin in Y. Then for any integer , the sets:

form a basis of neighborhoods of the origin for as i, l, and vary in all possible ways. If is a countable union of compact subsets and Y is a Fréchet space, then so is . Note that is convex whenever is convex. If Y is metrizable (resp. complete, locally convex, Hausdorff) then so is .[1][2] If is a basis of continuous seminorms for Y then a basis of continuous seminorms on is:

as i, l, and vary in all possible ways.[1]

If is a compact space and Y is a Banach space, then becomes a Banach space normed by .[2]

Space of Ck functions with support in a compact subset

We now duplicate the definition of the topology of the space of test functions. For any compact subset , let denote the set of all f in whose support lies in K (in particular, if then the domain of f is rather than K) and give the subspace topology induced by .[1] Let denote . Note that for any two compact subsets , the natural inclusion is an embedding of TVSs and that the union of all , as K varies over the compact subsets of , is .

Space of compactly support Ck functions

For any compact subset , let be the natural inclusion and give the strongest topology making all continuous. 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 natural injections .[1] 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 natural injections .[1] Each natural embedding is an embedding of TVSs. A subset S 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 canonical LF topology.

If Y is a Hausdorff locally convex space, T is a TVS, and is a linear map, then u is continuous if and only if for all compact , the restriction of u to is continuous.[1] One replace "all compact " with "all ".

Properties

Theorem[1] Let m be a positive integer and let be an open subset of . Given , for any let be defined by ; and let be defined by . Then is a (surjective) isomorphism of TVSs. Furthermore, the restriction is an isomorphism of TVSs when has its canonical LF topology.

Theorem[1] Let Y be a Hausdorff locally convex space. For every continuous linear form and every , let be defined by . Then is a continuous linear map; and furthermore, the restriction is also continuous (where has the canonical LF topology).

Identification as a tensor product

Suppose henceforth that Y is a Hausdorff space. 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 Y; this bilinear map turns this subspace into a tensor product of and Y, which we will denote by .[1] Furthermore, if denotes the vector subspace of consisting of all functions with compact support, then is a tensor product of and Y.[1]

If X is locally compact then is dense in while if X is an open subset of then is dense in .[2]

Theorem[2] If Y is a complete Hausdorff locally convex space, then is canonically isomorphic to the injective tensor product .

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See also

References

  1. Treves 2006, pp. 412-419.
  2. Treves 2006, pp. 446-451.
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