Spaces of test functions and distributions
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 formula_1 that have compact support. The space of all test functions, denoted by formula_2 is endowed with a certain topology, called the , that makes formula_3 into a complete Hausdorff locally convex TVS. The strong dual space of formula_3 is called and is denoted by formula_5 where the "formula_6" subscript indicates  that the continuous dual space of formula_2 denoted by formula_8 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 formula_9 then the use of Schwartz functions as test functions gives rise to a certain subspace of formula_10 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 formula_10 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 formula_12 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. Use of analytic test functions leads to Sato's theory of hyperfunctions. Notation. The following notation will be used throughout this article: Often, it is easier to just consider seminorms. Topology on "C"("K"). As before, fix formula_36 Recall that  if formula_37 is any compact subset of formula_14 then formula_39 For any compact subset formula_40 formula_41 is a closed subspace of the Fréchet space formula_42 and is thus also a Fréchet space. For all compact formula_43 satisfying formula_44 denote the inclusion map by formula_45 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 formula_41 is identical to the subspace topology it inherits from formula_47 and also formula_41 is a closed subset of formula_49  The interior of formula_50 relative to formula_51 is empty. If formula_17 is finite then formula_41 is a Banach space with a topology that can be defined by the norm formula_54 And when formula_55 then formula_56 is even a Hilbert space. The space formula_50 is a distinguished Schwartz Montel space so if formula_58 then it is normable and thus a Banach space (although like all other formula_59 it is a Fréchet space). Trivial extensions and independence of "C"("K")'s topology from "U". The definition of formula_41 depends on so we will let formula_61 denote the topological space formula_59 which by definition  is a topological subspace of formula_63 Suppose formula_64 is an open subset of formula_65 containing formula_14 and for any compact subset formula_67 let formula_68 is the vector subspace of formula_69 consisting of maps with support contained in formula_70 Given formula_71 its is by definition, the function formula_72 defined by: formula_73 so that formula_74 Let formula_75 denote the map that sends a function in formula_76 to its trivial extension on . This map is a linear injection and for every compact subset formula_77 (where formula_37 is also a compact subset of formula_64 since formula_80) we have formula_81 If is restricted to formula_82  then the following induced linear map is a homeomorphism (and thus a TVS-isomorphism): formula_83 and thus the next two maps (which like the previous map are defined by formula_84) are topological embeddings: formula_85 (the topology on formula_86 is the canonical LF topology, which is defined later). Using the injection formula_87 the vector space formula_76 is canonically identified with its image in formula_89 (however, if formula_90 then formula_91 is a topological embedding when these spaces are endowed with their canonical LF topologies, although it is continuous). Because formula_92 through this identification, formula_82 can also be considered as a subset of  formula_94 Importantly, the subspace topology formula_82 inherits from formula_42 (when it is viewed as a subset of formula_42) is identical to the subspace topology that it inherits from formula_69 (when formula_82 is viewed instead as a subset of formula_69 via the identification). Thus the topology on formula_61 is independent of the open subset of formula_65 that contains . This justifies the practice of written formula_41 instead of formula_104 Canonical LF topology. Recall that formula_76 denote all those functions in formula_42 that have compact support in formula_107 where note that formula_76 is the union of all formula_41 as ranges over formula_110 Moreover , for every , formula_76 is a dense subset of formula_63 The special case when formula_113 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 formula_114 if and only if formula_44 which in particular makes the collection formula_116 of compact subsets of into a directed set (we say that such a collection is ). For all compact formula_43 satisfying formula_44 there  are inclusion maps formula_119 Recall from above that the map formula_120 is a topological embedding. The collection of maps formula_121 forms a direct system in the category of locally convex topological vector spaces that is directed by formula_116 (under subset inclusion). This system's direct limit (in the category of locally convex TVSs) is the pair formula_123 where formula_124 are the natural inclusions and where formula_76 is now endowed with the (unique) strongest locally convex topology making all of the inclusion maps formula_126 continuous. Topology defined by neighborhoods of the origin. If is a convex subset of formula_127 then is  a neighborhood of the origin in the canonical LF topology if and only if it satisfies the following condition: Note that any convex set satisfying this condition is necessarily absorbing in formula_128 Since the topology of any topological vector space is translation-invariant, any TVS-topology is completely determined by the set of neighborhood of the origin. This means that one could actually the canonical LF topology by declaring that a convex balanced subset is a neighborhood of the origin if and only if it satisfies condition . Topology defined via differential operators. A is a sum formula_129 where formula_130 and  all but finitely many of formula_131 are identically . The integer formula_132 is called the of the differential operator formula_133 If formula_134 is a linear differential operator of order then it induces a canonical linear map formula_135 defined by formula_136 where we shall reuse notation and also denote this map by formula_133 For any formula_138 the canonical LF topology on formula_76 is the weakest locally convex TVS topology making all linear differential operators in formula_14 of order formula_141 into continuous maps from formula_76 into formula_143 Properties of the canonical LF topology. Canonical LF topology's independence from. One benefit of defining  the canonical LF topology as the direct limit of a direct system is that we may immediately use the universal property of direct limits. 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 formula_116 of compact sets. And by considering different collections formula_116 (in particular, those formula_116 mentioned at the beginning of this article), we may deduce different properties of this topology. In particular, we may deduce that the canonical LF topology makes formula_76 into a Hausdorff locally  convex strict LF-space (and also a strict LB-space if formula_148), which of course is the reason why this topology is called "the canonical LF topology" (see this footnote for more details). Universal property. From the universal property of direct limits, we know that if formula_149 is a linear map into a locally convex space (not necessarily Hausdorff), then is continuous if and only if is bounded if and only if for every formula_150 the restriction of to formula_41 is continuous (or bounded). Dependence of the canonical LF topology on. Suppose is an open subset of formula_65 containing formula_153  Let formula_154 denote the map that sends a function in formula_76 to its trivial extension on (which was defined above). This map is a continuous linear map. If (and only if) formula_90 then formula_157 is a dense subset of formula_158 and formula_91 is a topological embedding. Consequently, if formula_90 then the transpose of formula_91 is neither one-to-one nor onto. Bounded subsets. A subset formula_162 is bounded in formula_76 if and only if there exists some formula_164 such that formula_165 and formula_166 is a bounded subset of formula_167 Moreover, if formula_77 is compact and formula_169 then formula_170 is bounded  in formula_41 if and only if it is bounded in formula_63 For any formula_173 any bounded subset of formula_174 (resp. formula_175) is a relatively compact subset of formula_76 (resp. formula_42), where formula_178 Non-metrizability. For all compact formula_40 the interior of formula_41 in formula_76 is empty so that formula_76 is of the first category in itself. It follows from Baire's theorem that formula_76 is metrizable and thus also normable (see this footnote for an explanation of how the non-metrizable space formula_76 can be complete even though it does not admit a metric). The fact that formula_185 is a  nuclear Montel space makes up for the non-metrizability of formula_185 (see this footnote for a more detailed explanation). Relationships between spaces. Using the universal property of direct limits and the fact that the natural inclusions formula_120 are all topological embedding, one may show that all of the maps formula_188 are also topological embeddings. Said differently, the topology on formula_41 is identical to the subspace topology that it inherits from formula_127 where recall that formula_41's topology was to be the subspace topology induced on it by formula_63 In particular, both formula_76 and formula_42 induces the same subspace topology on  formula_167 However, this does ' imply that the canonical LF topology on formula_76 is equal to the subspace topology induced on formula_76 by formula_42; these two topologies on formula_76 are in fact ' equal to each other since the canonical LF topology is metrizable while the subspace topology induced on it by formula_42 is metrizable (since recall that formula_42 is metrizable). The canonical LF topology on formula_76 is actually than the subspace topology that it inherits from formula_42 (thus the natural inclusion formula_204 is continuous but a topological embedding). Indeed, the canonical LF topology is so fine that if formula_205 denotes some  linear map that is a "natural inclusion" (such as formula_206 or formula_207 or other maps discussed below) then this map will typically be continuous, which as is shown below, is ultimately the reason why locally integrable functions, Radon measures, etc. all induce distributions (via the transpose of such a "natural inclusion"). Said differently, the reason why there are so many different ways of defining distributions from other spaces ultimately stems from how very fine the canonical LF topology is. Moreover, since distributions are just continuous linear functionals on formula_12 the fine nature of the canonical LF topology means that more  linear functionals on formula_185 end up being continuous ("more" means as compared to a coarser topology that we could have placed on formula_185 such as for instance, the subspace topology induced by some formula_211 which although it would have made formula_185 metrizable, it would have also resulted in fewer linear functionals on formula_185 being continuous and thus there would have been fewer distributions; moreover, this particular coarser topology also has the disadvantage of not making formula_185 into a complete TVS). Distributions. As discussed earlier, continuous linear functionals on a formula_185 are known as distributions on . Thus the set of all  distributions on is the continuous dual space of formula_12 which when endowed with the strong dual topology is denoted by formula_220 We have the canonical duality pairing between a distribution on and a test function formula_221 which is denoted using angle brackets by formula_222 One interprets this notation as the distribution acting on the test function formula_19 to give a scalar, or symmetrically as the test function formula_19 acting on the distribution . Characterizations of distributions. Proposition. If is a linear functional on formula_185 then the following are equivalent: Topology on the space of distributions. The topology of uniform convergence on  bounded subsets is also called . This topology is chosen because it is with this topology that formula_10 becomes a nuclear Montel space and it is with this topology that the kernels theorem of Schwartz holds. No matter what dual topology is placed on formula_273 a of distributions converges in this topology if and only if it converges pointwise (although this need not be true of a net). No matter which topology is chosen, formula_10 will be a non-metrizable, locally convex topological vector space. The space formula_10 is separable and has the strong Pytkeev property but it is neither a  k-space nor a sequential space, which in particular implies that it is not metrizable and also that its topology can be defined using only sequences. Topological properties. Topological vector space categories. The canonical LF topology makes formula_76 into a complete distinguished strict LF-space (and a strict LB-space if and only if formula_148), which implies that formula_76 is a meager subset of itself. Furthermore, formula_127 as well as its strong dual space, is a complete Hausdorff locally convex barrelled bornological Mackey space. The strong dual of formula_76 is a Fréchet space if and only if formula_148 so in  particular, the strong dual of formula_12 which is the space formula_10 of distributions on , is metrizable (note that the weak-* topology on formula_10 also is not metrizable and moreover, it further lacks almost all of the nice properties that the strong dual topology gives formula_10). The three spaces formula_12 formula_287 and the Schwartz space formula_288 as well as the strong duals of each of these three spaces, are complete nuclear Montel bornological spaces, which implies that all six of these locally convex spaces are also paracompact reflexive barrelled Mackey spaces. The spaces formula_51 and formula_290 are both distinguished Fréchet spaces . Moreover, both formula_185 and formula_290 are Schwartz TVSs. Convergent sequences. Convergent sequences and their insufficiency to describe topologies. The strong dual spaces of formula_51 and formula_290 are sequential spaces but not Fréchet-Urysohn spaces. Moreover, neither the space of test functions formula_185 nor its strong dual formula_10 is a sequential space (not even an Ascoli space), which in particular implies that their topologies can be defined entirely in terms of convergent sequences. A sequence formula_226 in formula_76 converges in formula_76 if and only if there exists some formula_164 such that formula_41 contains this sequence and this sequence converges in formula_41 ; equivalently, it converges if and only if the following two conditions hold: Neither the space formula_185 nor its strong dual formula_10 is a sequential space, and consequently, their topologies can be defined entirely in terms of convergent sequences. For this reason, the above characterization of when a sequence converges is enough to define the canonical LF topology on formula_245 The same can be said of the strong dual topology on formula_220 What sequences do characterize. Nevertheless, sequences do characterize many important properties, as we now discuss. It is known that in the dual space of any Montel space, a sequence  converges in the strong dual topology if and only if it converges in the weak* topology, which in particular, is the reason why a sequence of distributions converges (in the strong dual topology) if and only if it converges pointwise (this leads many authors to use pointwise convergence to actually the convergence of a sequence of distributions; this is fine for sequences but it does extend to the convergence of nets of distributions since a net may converge pointwise but fail to converge in the strong dual topology). Sequences characterize continuity of linear maps valued in locally convex space. Suppose  is a locally convex bornological space (such as any of the six TVSs mentioned earlier). Then a linear map formula_312 into a locally convex space is continuous if and only if it maps null sequences in to bounded subsets of . More generally, such a linear map formula_312 is continuous if and only if it maps Mackey convergent null sequences to bounded subsets of formula_314 So in particular, if a linear map formula_312 into a locally convex space is sequentially continuous at the origin then it is continuous. However, this does necessarily extend to non-linear maps and/or to maps valued  in topological spaces that are not locally convex TVSs. For every formula_316 is sequentially dense in formula_128 Furthermore, formula_318 is a sequentially dense subset of formula_10 (with its strong dual topology) and also a sequentially dense subset of the strong dual space of formula_320 Sequences of distributions. A sequence of distributions formula_321 converges with respect to the weak-* topology on formula_10 to a distribution if and only if formula_323 for every test function formula_324 For example, if formula_325 is the function formula_326 and formula_327 is the distribution corresponding to formula_328 then formula_329 as formula_330 so formula_331 in formula_332 Thus, for  large formula_333 the function formula_334 can be regarded as an approximation of the Dirac delta distribution. Tensor product of distributions. Let formula_345 and formula_346 be open sets. Assume all vector spaces to be over the field formula_347 where formula_348 or formula_349 For formula_350 define for every formula_351 and every formula_352 the following functions: formula_353 Given formula_354 and formula_355 define the following functions: formula_356 where formula_357 and formula_358 These definitions associate every formula_359 and formula_360 with the (respective) continuous linear map: formula_361 Moreover if either formula_170 (resp. formula_363) has compact support then it also induces a continuous linear map of formula_364  (resp. formula_368 Schwartz kernel theorem. The tensor product defines a bilinear map formula_369 the span of the range of this map is a dense subspace of its codomain. Furthermore, formula_370 Moreover formula_371 induces continuous bilinear maps: formula_372 where formula_373 denotes the space of distributions with compact support and formula_374 is the Schwartz space of rapidly decreasing functions. This result does not hold for Hilbert spaces such as formula_375 and its dual space. Why does such a result hold for the space of distributions and test functions but not for other "nice" spaces like the Hilbert space formula_375? This question led  Alexander Grothendieck to discover nuclear spaces, nuclear maps, and the injective tensor product. He ultimately showed that it is precisely because formula_377 is a nuclear space that the Schwartz kernel theorem holds. Like Hilbert spaces, nuclear spaces may be thought as of generalizations of finite dimensional Euclidean space. Spaces of distributions. For all formula_378 and all formula_379 all of the following canonical injections are continuous and have an image (i.e. range) that is a dense subset of their codomain: formula_380 where the topologies on formula_381 (formula_382) are defined as direct limits of the spaces formula_383 in a manner analogous to  how the canonical LF-topologies on formula_76 were defined (so in particular, they are not the usual norm topologies). The range of each of the maps above (and of any composition of the maps above) is dense in the codomain. Indeed, formula_185 is even sequentially dense in every formula_128 All of the canonical injections formula_387 (formula_388) are continuous and the range of this injection is dense in the codomain if and only if formula_389 (here formula_390 has its usual norm topology). Suppose that formula_391 is one of the spaces formula_76 (for formula_393) or formula_394 (for formula_388) or formula_390 (for formula_397 ). Because the canonical injection formula_398 is a continuous injection whose image is dense in the codomain, this map's transpose formula_399 is a continuous injection. This injective transpose map thus allows the continuous dual space formula_400 of formula_391 to be identified with a certain vector subspace of the space formula_402 of all distributions (specifically, it is identified with the image of this transpose map). This continuous transpose map is not necessarily a TVS-embedding so the topology that this map transfers from its domain to the image formula_403 is finer than the subspace topology that this space inherits from formula_220  A linear subspace of formula_10 carrying a locally convex topology that is finer than the subspace topology induced by formula_406 is called . Almost all of the spaces of distributions mentioned in this article arise in this way (e.g. tempered distribution, restrictions, distributions of order formula_407 some integer, distributions induced by a positive Radon measure, distributions induced by an formula_408-function, etc.) and any representation theorem about the dual space of may, through the transpose formula_409 be transferred directly to elements of the space formula_410 Radon measures. The inclusion map formula_411 is a continuous injection whose image is dense in its  codomain, so the transpose formula_412 is also a continuous injection. Note that the continuous dual space formula_413 can be identified as the space of Radon measures, where there is a one-to-one correspondence between the continuous linear functionals formula_414 and integral with respect to a Radon measure; that is, Through the injection formula_421 every Radon measure becomes a distribution on . If formula_19 is a locally integrable function on then the distribution formula_423 is a Radon measure; so Radon measures form a large and important space of distributions. The following is the theorem of the structure of distributions of Radon  measures, which shows that every Radon measure can be written as a sum of derivatives of locally formula_424 functions in : Positive Radon measures A linear function on a space of functions is called if whenever a function formula_19 that belongs to the domain of is non-negative (meaning that formula_19 is real-valued and formula_427) then formula_428 One may show that every positive linear functional on formula_419 is necessarily continuous (that is, necessarily a Radon measure). Lebesgue measure is an example of a positive Radon measure. Locally integrable functions as distributions. One particularly important class of Radon measures are those  that are induced locally integrable functions. The function formula_430 is called if it is Lebesgue integrable over every compact subset of . This is a large class of functions which includes all continuous functions and all Lp space formula_408 functions. The topology on formula_377 is defined in such a fashion that any locally integrable function formula_19 yields a continuous linear functional on formula_377 – that is, an element of formula_10 – denoted here by formula_436, whose value on the test function formula_437 is given by the Lebesgue integral: formula_438 Conventionally, one abuses notation by identifying formula_436 with formula_21 provided no confusion can arise , and thus the pairing between formula_436 and formula_437 is often written formula_443 If formula_19 and are two locally integrable functions, then the associated distributions formula_436 and are equal to the same element of formula_10 if and only if formula_19 and are equal almost everywhere (see, for instance, ). In a similar manner, every Radon measure formula_416 on defines an element of formula_10 whose value on the test function formula_437 is formula_451 As above, it is conventional to abuse notation and write the pairing between a Radon measure formula_416 and a test function formula_437 as formula_454 Conversely, as shown in a theorem  by Schwartz (similar to the Riesz representation theorem), every distribution which is non-negative on non-negative functions is of this form for some (positive) Radon measure. Test functions as distributions The test functions are themselves locally integrable, and so define distributions. The space of test functions formula_185 is sequentially dense in formula_10 with respect to the strong topology on formula_220 This means that for any formula_458 there is a sequence of test functions, formula_459 that converges to formula_460 (in its strong dual topology) when considered as a sequence of distributions. Or equivalently, formula_461 Furthermore, formula_185 is also sequentially dense  in the strong dual space of formula_320 Distributions with compact support. The inclusion map formula_464 is a continuous injection whose image is dense in its codomain, so the transpose formula_465 is also a continuous injection. Thus the image of the transpose, denoted by formula_466 forms a space of distributions when it is endowed with the strong dual topology of formula_467 (transferred to it via the transpose map formula_468 so the topology of formula_469 is finer than the subspace topology that this set inherits from formula_10). The elements of formula_471 can be identified as the space of distributions with compact support . Explicitly, if is a distribution on then the following are equivalent, Compactly supported distributions define continuous linear functionals on the space formula_51; recall that the topology on formula_51 is defined such that a sequence of test functions formula_480 converges to 0 if and only if all derivatives of formula_480 converge uniformly to 0 on every compact subset of . Conversely, it can be shown that every continuous linear functional on this space defines a distribution of compact support. Thus compactly supported distributions can be identified with those distributions that can be extended from formula_185 to formula_320 Distributions of finite order. Let  formula_484 The inclusion map formula_485 is a continuous injection whose image is dense in its codomain, so the transpose formula_486 is also a continuous injection. Consequently, the image of formula_487 denoted by formula_488 forms a space of distributions when it is endowed with the strong dual topology of formula_489 (transferred to it via the transpose map formula_490 so formula_491's topology is finer than the subspace topology that this set inherits from formula_10). The elements of formula_493 are ' The distributions of order formula_494 which are also called ' are exactly the distributions that are Radon measures (described above). For formula_495 a  is a distribution of order formula_496 that is not a distribution of order formula_497 A distribution is said to be of if there is some integer such that it is a distribution of order formula_498 and the set of distributions of finite order is denoted by formula_499 Note that if formula_500 then formula_501 so that formula_502 is a vector subspace of formula_10 and furthermore, if and only if formula_504 Every distribution with compact support in is a distribution of finite order. Indeed, every distribution in is a distribution of finite order, in the following sense: If is an open and  relatively compact subset of and if formula_505 is the restriction mapping from to , then the image of formula_10 under formula_505 is contained in formula_508 The following is the theorem of the structure of distributions of finite order, which shows that every distribution of finite order can be written as a sum of derivatives of Radon measures: Example. (Distributions of infinite order) Let formula_509 and for every test function formula_21 let formula_511 Then is a distribution of infinite order on . Moreover, can not be extended to a distribution on formula_512; that is, there exists no distribution on formula_512 such that the  restriction of to is equal to . Tempered distributions and Fourier transform. Defined below are the , which form a subspace of formula_514 the space of distributions on formula_515 This is a proper subspace: while every tempered distribution is a distribution and an element of formula_514 the converse is not true. Tempered distributions are useful if one studies the Fourier transform since all tempered distributions have a Fourier transform, which is not true for an arbitrary distribution in formula_517 The Schwartz space, formula_288 is the space of all smooth functions that are rapidly decreasing at infinity along with all partial derivatives. Thus  formula_519 is in the Schwartz space provided that any derivative of formula_520 multiplied with any power of formula_521 converges to 0 as formula_522 These functions form a complete TVS with a suitably defined family of seminorms. More precisely, for any multi-indices formula_523 and formula_524 define: formula_525 Then formula_437 is in the Schwartz space if all the values satisfy: formula_527 The family of seminorms formula_528 defines a locally convex topology on the Schwartz space. For formula_529 the seminorms are, in fact, norms on the Schwartz space. One can also use the following family of seminorms to define the topology: formula_530  Otherwise, one can define a norm on formula_290 via formula_532 The Schwartz space is a Fréchet space (i.e. a complete metrizable locally convex space). Because the Fourier transform changes formula_533 into multiplication by formula_534 and vice versa, this symmetry implies that the Fourier transform of a Schwartz function is also a Schwartz function. A sequence formula_535 in formula_290 converges to 0 in formula_290 if and only if the functions formula_538 converge to 0 uniformly in the whole of formula_539 which implies that such a sequence must converge to zero in formula_540 formula_541 is dense in formula_542 The subset of all  analytic Schwartz functions is dense in formula_290 as well. The Schwartz space is nuclear and the tensor product of two maps induces a canonical surjective TVS-isomorphisms formula_544 where formula_545 represents the completion of the injective tensor product (which in this case is the identical to the completion of the projective tensor product). The inclusion map formula_546 is a continuous injection whose image is dense in its codomain, so the transpose formula_547 is also a continuous injection. Thus, the image of the transpose map, denoted by formula_548 forms a space of distributions when it is endowed with the strong dual  topology of formula_549 (transferred to it via the transpose map formula_550 so the topology of formula_551 is finer than the subspace topology that this set inherits from formula_552). The space formula_551 is called the space of . It is the continuous dual of the Schwartz space. Equivalently, a distribution is a tempered distribution if and only if formula_554 The derivative of a tempered distribution is again a tempered distribution. Tempered distributions generalize the bounded (or slow-growing) locally integrable functions; all distributions with compact support and all square-integrable functions are tempered distributions. More generally, all functions that are products of  polynomials with elements of Lp space formula_555 for formula_556 are tempered distributions. The can also be characterized as , meaning that each derivative of grows at most as fast as some polynomial. This characterization is dual to the behaviour of the derivatives of a function in the Schwartz space, where each derivative of formula_437 decays faster than every inverse power of formula_558 An example of a rapidly falling function is formula_559 for any positive formula_560 To study the Fourier transform, it is best to consider complex-valued test functions and complex-linear distributions. The ordinary continuous Fourier transform formula_561 is a  TVS-automorphism of the Schwartz space, and the is defined to be its transpose formula_562 which (abusing notation) will again be denoted by . So the Fourier transform of the tempered distribution is defined by formula_563 for every Schwartz function formula_564 formula_565 is thus again a tempered distribution. The Fourier transform is a TVS isomorphism from the space of tempered distributions onto itself. This operation is compatible with differentiation in the sense that formula_566 and also with convolution: if is a tempered distribution and formula_567 is a smooth function on formula_539 formula_569 is again a tempered distribution and formula_570 is the  convolution of formula_565 and formula_572. In particular, the Fourier transform of the constant function equal to 1 is the formula_573 distribution. If formula_574 is a tempered distribution, then there exists a constant formula_249 and positive integers and such that for all Schwartz functions formula_576 formula_577 This estimate along with some techniques from functional analysis can be used to show that there is a continuous slowly increasing function and a multi-index formula_523 such that formula_579 Restriction of distributions to compact sets If formula_580 then for any compact set formula_581 there exists a continuous function compactly supported in formula_65 (possibly on  a larger set than itself) and a multi-index formula_523 such that formula_584 on formula_585 Using holomorphic functions as test functions. The success of the theory led to investigation of the idea of hyperfunction, in which spaces of holomorphic functions are used as test functions. A refined theory has been developed, in particular Mikio Sato's algebraic analysis, using sheaf theory and several complex variables. This extends the range of symbolic methods that can be made into rigorous mathematics, for example Feynman integrals.