In mathematical analysis, the spaces of test functions and distributions are topological vector spaces (TVSs) that are used in the definition and application of distributions. Test functions are usually infinitely differentiable complex-valued (or sometimes real-valued) functions on a non-empty open subset that have compact support. The space of all test functions, denoted by , is endowed with a certain topology, called the , that makes into a complete Hausdorff locally convex TVS. The strong dual space of is called and is denoted by where the "" subscript indicates that the continuous dual space of denoted by is endowed with the strong dual topology.
There are other possible choices for the space of test functions, which lead to other different spaces of distributions. If then the use of Schwartz functionsThe Schwartz space consists of smooth rapidly decreasing test functions, where "rapidly decreasing" means that the function decreases faster than any polynomial increases as points in its domain move away from the origin. as test functions gives rise to a certain subspace of whose elements are called . These are important because they allow the Fourier transform to be extended from "standard functions" to tempered distributions. The set of tempered distributions forms a vector subspace of the space of distributions and is thus one example of a space of distributions; there are many other spaces of distributions.
There also exist other major classes of test functions that are subsets of , such as spaces of analytic test functions, which produce very different classes of distributions. The theory of such distributions has a different character from the previous one because there are no analytic functions with non-empty compact support.Except for the trivial (i.e. identically ) map, which of course is always analytic. Use of analytic test functions leads to Sato's theory of hyperfunctions.
Notation
The following notation will be used throughout this article: is a fixed positive integer and is a fixed non-empty open subset of Euclidean space denotes the natural numbers. will denote a non-negative integer or If is a function then will denote its domain and the of denoted by is defined to be the closure of the set in For two functions , the following notation defines a canonical pairing: A of size is an element in (given that is fixed, if the size of multi-indices is omitted then the size should be assumed to be ). The of a multi-index is defined as and denoted by Multi-indices are particularly useful when dealing with functions of several variables, in particular we introduce the following notations for a given multi-index : We also introduce a partial order of all multi-indices by if and only if for all When we define their multi-index binomial coefficient as: will denote a certain non-empty collection of compact subsets of (described in detail below).
Definitions of test functions and distributions
In this section, we will formally define real-valued distributions on . With minor modifications, one can also define complex-valued distributions, and one can replace with any (paracompact) smooth manifold.
Note that for all and any compact subsets and of , we have:
Distributions on are defined to be the continuous linear functionals on when this vector space is endowed with a particular topology called the . This topology is unfortunately not easy to define but it is nevertheless still possible to characterize distributions in a way so that no mention of the canonical LF-topology is made.
Proposition: If is a linear functional on then the is a distribution if and only if the following equivalent conditions are satisfied: For every compact subset there exist constants and (dependent on ) such that for all For every compact subset there exist constants and such that for all with support contained in See for example . For any compact subset and any sequence in if converges uniformly to zero on for all multi-indices , then
The above characterizations can be used to determine whether or not a linear functional is a distribution, but more advanced uses of distributions and test functions (such as applications to differential equations) is limited if no topologies are placed on and . To define the space of distributions we must first define the canonical LF-topology, which in turn requires that several other locally convex topological vector spaces (TVSs) be defined first. First, a (non-normable) topology on will be defined, then every will be endowed with the subspace topology induced on it by , and finally the (non-metrizable) canonical LF-topology on will be defined. The space of distributions, being defined as the continuous dual space of , is then endowed with the (non-metrizable) strong dual topology induced by and the canonical LF-topology (this topology is a generalization of the usual operator norm induced topology that is placed on the continuous dual spaces of normed spaces). This finally permits consideration of more advanced notions such as convergence of distributions (both sequences nets), various (sub)spaces of distributions, and operations on distributions, including extending differential equations to distributions.
Choice of compact sets K
Throughout, will be any collection of compact subsets of such that (1) and (2) for any compact there exists some such that The most common choices for are: The set of all compact subsets of , or A set where and for all , and is a relatively compact non-empty open subset of (here, "relatively compact" means that the closure of , in either or , is compact).
We make into a directed set by defining if and only if Note that although the definitions of the subsequently defined topologies explicitly reference , in reality they do not depend on the choice of that is, if and are any two such collections of compact subsets of then the topologies defined on and by using in place of are the same as those defined by using in place of
Topology on Ck(U)
We now introduce the seminorms that will define the topology on Different authors sometimes use different families of seminorms so we list the most common families below. However, the resulting topology is the same no matter which family is used.
All of the functions above are non-negative -valuedThe image of the compact set under a continuous -valued map (for example, for ) is itself a compact, and thus bounded, subset of If then this implies that each of the functions defined above is -valued (that is, none of the supremums above are ever equal to ). seminorms on As explained in this article, every set of seminorms on a vector space induces a locally convex vector topology.
Each of the following sets of seminorms
generate the same locally convex vector topology on .
With this topology, becomes a locally convex Fréchet space that is normable. Every element of is a continuous seminorm on . Under this topology, a net in converges to if and only if for every multi-index with and every compact , the net of partial derivatives converges uniformly to on For any any (von Neumann) bounded subset of is a relatively compact subset of In particular, a subset of is bounded if and only if it is bounded in for all The space is a Montel space if and only if
The topology on is the superior limit of the subspace topologies induced on by the TVSs as ranges over the non-negative integers. A subset of is open in this topology if and only if there exists such that is open when is endowed with the subspace topology induced on it by .
Metric defining the topology
If the family of compact sets satisfies and for all then a complete translation-invariant metric on can be obtained by taking a suitable countable Fréchet combination of any one of the above defining families of seminorms (A, B, C). For example, using the seminorms results in the metric
Often, it is easier to just consider seminorms (avoiding any metric) and use the tools of functional analysis.
Topology on Ck(K)
As before, fix Recall that if is any compact subset of then
For any compact subset is a closed subspace of the Fréchet space and is thus also a Fréchet space. For all compact satisfying denote the inclusion map by Then this map is a linear embedding of TVSs (that is, it is a linear map that is also a topological embedding) whose image (or "range") is closed in its codomain; said differently, the topology on is identical to the subspace topology it inherits from and also is a closed subset of The interior of relative to is empty.
If is finite then is a Banach space with a topology that can be defined by the norm
And when then is even a Hilbert space. The space is a distinguished Schwartz Montel space so if then it is normable and thus a Banach space (although like all other it is a Fréchet space).
Trivial extensions and independence of Ck(K)'s topology from U
The definition of depends on so we will let denote the topological space which by definition is a topological subspace of Suppose is an open subset of containing and for any compact subset , let is the vector subspace of consisting of maps with support contained in Given , its is by definition, the function defined by:
so that Let denote the map that sends a function in to its trivial extension on . This map is a linear injection and for every compact subset (where is also a compact subset of since ) we have
If is restricted to then the following induced linear map is a homeomorphism (and thus a TVS-isomorphism):
and thus the next two maps (which like the previous map are defined by ) are topological embeddings:
(the topology on is the canonical LF topology, which is defined later). Using the injection
the vector space is canonically identified with its image in (however, if then is a topological embedding when these spaces are endowed with their canonical LF topologies, although it is continuous). Because , through this identification, can also be considered as a subset of Importantly, the subspace topology inherits from (when it is viewed as a subset of ) is identical to the subspace topology that it inherits from (when is viewed instead as a subset of via the identification). Thus the topology on is independent of the open subset of that contains . This justifies the practice of writing instead of .
Canonical LF topology
Recall that denote all those functions in that have compact support in where note that is the union of all as ranges over Moreover, for every , is a dense subset of The special case when gives us the space of test functions.
This section defines the canonical LF topology as a direct limit. It is also possible to define this topology in terms of its neighborhoods of the origin, which is described afterwards.
Topology defined by direct limits
For any two sets and , we declare that if and only if , which in particular makes the collection of compact subsets of into a directed set (we say that such a collection is ). For all compact satisfying there are inclusion maps
Recall from above that the map is a topological embedding. The collection of maps forms a direct system in the category of locally convex topological vector spaces that is directed by (under subset inclusion). This system's direct limit (in the category of locally convex TVSs) is the pair where are the natural inclusions and where is now endowed with the (unique) strongest locally convex topology making all of the inclusion maps continuous.
Topology defined by neighborhoods of the origin
If is a convex subset of , then is a neighborhood of the origin in the canonical LF topology if and only if it satisfies the following condition: