Spaces of test functions and distributions

These are important because they allow the Fourier transform to be extended from "standard functions" to tempered distributions.

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.

[note 2] Use of analytic test functions leads to Sato's theory of hyperfunctions.

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.

can be obtained by taking a suitable countable Fréchet combination of any one of the above defining families of seminorms (A through D).

Often, it is easier to just consider seminorms (avoiding any metric) and use the tools of functional analysis.

is now endowed with the (unique) strongest locally convex topology making all of the inclusion maps

This means that one could actually define the canonical LF topology by declaring that a convex balanced subset U is a neighborhood of the origin if and only if it satisfies condition CN.

is the weakest locally convex TVS topology making all linear differential operators in

Another benefit is that we can use well-known results from category theory to deduce that the canonical LF topology is actually independent of the particular choice of the directed collection

is a linear map into a locally convex space Y (not necessarily Hausdorff), then u is continuous if and only if u is bounded if and only if for every

[note 7] Using the universal property of direct limits and the fact that the natural inclusions

the fine nature of the canonical LF topology means that more linear functionals on

becomes a nuclear Montel space and it is with this topology that the kernels theorem of Schwartz holds.

is separable[16] and has the strong Pytkeev property[17] but it is neither a k-space[17] nor a sequential space,[16] which in particular implies that it is not metrizable and also that its topology can not be defined using only sequences.

also is not metrizable and moreover, it further lacks almost all of the nice properties that the strong dual topology gives

For this reason, the above characterization of when a sequence converges is not enough to define the canonical LF topology on

Sequences characterize continuity of linear maps valued in locally convex space.

Suppose X is a locally convex bornological space (such as any of the six TVSs mentioned earlier).

into a locally convex space Y is continuous if and only if it maps null sequences[note 9] in X to bounded subsets of Y.

is continuous if and only if it maps Mackey convergent null sequences[note 10] to bounded subsets of

This is because the transpose allows for a unified presentation of the many definitions in the theory of distributions and also because its properties are well known in functional analysis.

can be identified as the space of Radon measures, where there is a one-to-one correspondence between the continuous linear functionals

is the space of all smooth functions that are rapidly decreasing at infinity along with all partial derivatives.

These functions form a complete TVS with a suitably defined family of seminorms.

[39] The Schwartz space is nuclear and the tensor product of two maps induces a canonical surjective TVS-isomorphisms

which (abusing notation) will again be denoted by F. So the Fourier transform of the tempered distribution T is defined by

[14] Schwartz kernel theorem[40] — Each of the canonical maps below (defined in the natural way) are TVS isomorphisms:

This extends the range of symbolic methods that can be made into rigorous mathematics, for example Feynman integrals.