Infinite-Dimensional Linear Algebra and Solvability of Partial Differential Equations

We discuss linear algebra of infinite-dimensional vector spaces in terms of algebraic (Hamel) bases. As an application we prove the surjectivity of a large class of linear partial differential operators with smooth ($\mathcal C^\infty$-coefficients) coefficients, called in the article \emph{regular}, acting on the algebraic dual $\mathcal D^*(\Omega)$ of the space of test-functions $\mathcal D(\Omega)$. The surjectivity of the partial differential operators guarantees solvability of the corresponding partial differential equations within $\mathcal D^*(\Omega)$. We discuss our result in contrast to and comparison with similar results about the restrictions of the regular operators on the space of Schwartz distribution $\mathcal D^\prime(\Omega)$, where these operators are often non-surjective.


Introduction
In Section 3-5 we present the basic results of infinite-dimensional linear algebra, an old branch of mathematics initiated in1905 by Georg Hamel [12], dealing with infinitedimensional vector spaces in terms of algebraic (Hamel) bases rather than topological or orthonormal Hilbert bases. The approach is mostly algebraic. In Theorem 5.4 we show that a linear operator is injective if and only if its dual operator is surjective; a result well-known for finite-dimensional vector spaces but less-known for infinitedimensional spaces. This gives rise to the Definition 5.5 of a regular linear operatora surjective operator on the dual space with injective co-dual.
Several discussions of the earlier versions of this text convinced us that the algebraic (Hamel) bases have gradually been falling out of popularity in the last several decades.
That is why the first part of the article (Section 3-5) is written in somewhat tutorial manner, with many illustrative examples (Section 7). A reader who knows Theorem 5.4 from the finite-dimensional linear algebra and who believes in its validity for infinitedimensional vector spaces might skip reading the first several sections and start directly from Section 8.
In Sections 8 we apply infinite-dimensional linear algebra to the particular case of the vector space D(Ω) and its algebraic dual D * (Ω). Here Ω is an open set of R d in the usual topology of R d . Somewhere in this section we abandon the realm of algebra and start involving concepts and methods from functional analysis and the theory of partial differential operators (Hörmander [15]- [17]). In particular, Definition 5.5 (mentioned above) -if applied to D(Ω) -gives rise to the concept of regular operator with C ∞coefficients: a surjective linear operator P * (x, ∂) with C ∞ -coefficients acting on D * (Ω) which has an injective co-dual (transposed) operator P(x, ∂) on D(Ω). (Vladimirov [39]) who are otherwise interested in the main topic of our article, we offer a characterization of the space of Schwartz distributions D ′ (Ω) as a particular subspace of D * (Ω) without the usual involvement of the strong topology on the space of test-functions D(Ω) (Section 9). We shortly outline a sequential approach to distribution theory based on our characterization (Remark 9.2). Thus, the dilemma D ′ (Ω) vs. D * (Ω) -discussed in Section 11 -can be followed by readers without strong (or any) background in Schwartz theory of distributions.

For readers without background in Schwartz theory of distributions
In Section 11 we identify several subclasses of linear partial differential operators in mathematics (Hörmander [15]- [17]) as regular (thus, surjective on D * (Ω)) which include the following: • All linear partial differential operators with constant coefficients are regular.
• All elliptic operators with analytic coefficients are regular.
All of these operators are surjective on D * (Ω), but not necessarily surjective on the following three invariant subspaces D(Ω), E(Ω) and D ′ (Ω) (Section 10). Consequently, we prove the solvability of the partial differential equations of the form P * (x, ∂)U = T in D * (Ω), for regular operators P * (x, ∂) : D * (Ω) → D * (Ω). In other words, we prove the existence of a solution U in D * (Ω) for every choice of T also in D * (Ω). We should recall that: • Every linear partial differential operator with constant coefficients P * (∂) on D ′ (R d ) is surjective; this is the famous existence theorem of Malgrange [28] and Ehrenpreis [9].
• The Malgrange-Ehrenpreis existence result might, however, fail in D ′ (Ω) for operators which are hypoelliptic but not elliptic, subsets Ω of R d which are open, but not P-convex for supports (Hörmander [14], Theorem 10.6.6, Corollary 10.6.8). Thus, the partial differential equation P * (∂)U = f might have no solutions in D ′ (Ω) even for some smooth f .
• Hans Lewy [23] was the first to show that the Lewy operator L * (x, ∂) is not surjective on D ′ (R 3 ). Thus, the partial differential equations of the form L * (x, ∂)U = ϕ might fail to have solution U in D ′ (R 3 ) even for some testfunctions ϕ ∈ D(R 3 ). A general existence result also fails in the space of hyperfunctions (Schapira [35]).
• The elliptic operators mentioned above are, in general, also non-surjective on D ′ (Ω).
In Section 12 we show -with the help of Hamel bases -that the space of generalized distributions E(Ω) introduced in (Todorov [37], §2) can be embedded as a C-vector subspace into the algebraic dual D * (Ω) of the space of test-functions D(Ω). Because E(Ω) was defined in the framework of non-standard analysis (Robinson [33]), we look upon D * (Ω) as a standardization of E(Ω). Actually, our article itself can be viewed as a standardization of the results in Todorov [37], because the surjectivity of the regular operators was first proved in Todorov [37] in the framework of L( E(Ω)), while the main result of this article (Theorem 8.4) holds within L(D * (Ω)). Thus, by replacing E(Ω) with D * (Ω), our result about the regular operators becomes accessible even for readers without background in non-standard analysis. Our standardization is, of course, not an isolated event in mathematics; we remind two more cases of standardizations in the history of mathematics.
Our inspiration comes from the fundamental theorem of algebra: following this analogy the space D ′ (Ω) is the counterpart of the field of real numbers R, the space D * (Ω) is the counterpart of the field of complex numbers C, and the class of regular operator is the counterpart of the ring of polynomials C[x]. We are trying to convince the reader that the space D * (Ω) -rather than D ′ (Ω) -deserves to be considered as the natural theoretical framework for the class of regular operators P * (x, ∂), since the equations of the form P * (x, ∂)U = T often have no solutions in D ′ (Ω).
Recall as well that the global solvability of arbitrary analytic partial differential equations was studied in (Rosinger [34], Chapter 2) and (Oberguggenberger [29], Section 22). The existence results for continuous partial differential operators are obtained by means of the Dedekind order completion method in Oberguggenberger & Rosinger [31].
As we mentioned above, the surjectivity of the regular operators was first proved in Todorov [37] in the framework of non-standard analysis. Meanwhile (in the period between the publication of Todorov [37] and the writing of this article) two more similar articles in the framework of D * (Ω) appeared: an unpublished manuscript Oberguggenberger & Todorov [32] and Oberguggen-berger [30]. In this article we shall use some of the results in Oberguggenberger [30].
On the topic of solvability of differential equations we refer to a relatively recent survey in Dencker [?] (no connection with the space D * (Ω)).
Finally, we should mention that our article has somewhat an ideological edge because we challenge at least two widely spread prejudices in the mathematical community: The first one is that Hamel bases are not and can never be mathematically useful.
The second one is that we should never go beyond the space of Schwartz distributions D ′ (Ω) as a framework of a partial differential equation, especially if the equation is linear. That is to say "better to admit (perhaps with some regret) that a given equation has no solutions rather than look for a solution outside D ′ (Ω)".

Notations and Set-Theoretical Framework
The set-theoretical framework of this text is the usual ZFC-axioms (Zermelo-Fraenkel axioms with the axiom of choice) along with the GCH (Generalized Continuum Hypothesis) in the form 2 κ = κ + for every cardinal κ (or equivalently, 2 ℵα = ℵ α+1 for all ordinals α). Here we write κ + for the successor of κ. For the domain of ZFC and GCH axioms we use the superstructures S with the set of individuals S = K ∪ V , where V is the vector space over a field K under consideration (e.g. V = R n with K = R or V = C n with K = C, etc.). Our formal language is based on bounded quantifiers of the form (∀x ∈ A)α(x) and (∃x ∈ B)β(x), where A, B ∈ S \ S and α(x) and β(x) are predicates (Davis [8], p.11-15). We believe however, that the rest of this text can be followed without a familiarity with the concept of superstructure.
• Every set can be well-ordered.
• The usual partial order on the class of cardinal numbers is a total order.
In particular, Zorn's Lemma will be involved in Theorem 3.6 and the total order between cardinals is needed in the proof of Lemma 3.9. Also, "every set can be well-ordered" will be useful to supply a basis with well-ordering if desired (Remark 6.4).
Actually, we do not need the GCH except for the purpose of simplifying the calculations with cardinals and the dimension of vector spaces. For example, with the help of GCH, If X is a set, we shall treat X as a subset of the power set P(X), in symbols, X ⊂ P(X) by means of the embedding x → {x}. If X and Y are two sets, we denote by Y X the set of all functions from X to Y .
For index sets (for indexing bases, for example) we use the popular sets: N, R, R d , P(R), P(R d ), P(P(R)), etc. with cardinalities ℵ 0 , c, c, c + , c + , (c + ) + , respectively. We sometimes use the field of scalars K itself as an index set or K d , P(K), P(K d ), P(P(K)), etc.
In what follows V stands for a generic vector space over a field K (Axler [2]).
Sometimes we shall write V| K instead of V . If we write U ⊆ V , we mean that both U and V are vector spaces over the same field and U is a vector subspace of V . Similarly, V ∼ = W means that V and W are isomorphic vector spaces. L(V) denotes the K-vector space consisting of all linear operators L : V → V . We denote by V * the algebraic dual of V . We denote by Let T d denote the usual topology on R d and let X, Y ∈ T d be two open set of R d . We denote by Diff(X, Y) the set of all diffeomorphisms from X to Y . If θ ∈ Diff(X, Y), we denote by J θ : X → R, J θ = | det ∂θ ∂x |, the corresponding Jacobian determinant. We denote by Diff(X) the group of diffeomorphisms from X to itself.
Let Ω stand for a (generic) open subset of R d . Here is a list of popular functional spaces and notations: • E(Ω) = C ∞ (Ω) denotes the space C ∞ -functions from Ω to C.
• L 2 (Ω) denotes the usual Hilbert space of Lebesgue measurable square integrable functions from Ω to C.
• L ∞ (Ω) denotes the space of Lebesgue measurable bounded functions from Ω to C.
• L loc (Ω) stands for the Lebesgue measurable locally integrable functions from Ω to C.
• E ′ (Ω) denotes the space of Schwartz distributions with compact support.
Definition 3.1 (Basis and Dimension) Let V be a non-trivial vector space over a field K.

1.
A subset B of V is called free if every finite subset of B consists of linearly independent vectors in V .

2.
A free set B of V is called maximal (or, a maximal free set) if B cannot be extended (properly) to a free set of V . Every maximal free set B of V is called a algebraic basis, Hamel basis or simply, basis of V .

Corollary 3.5 (Two Equalities)
(i) If card K < card V , then dim V = card V (see Example 7.1 in this paper).
(ii) If dim V < card V , then card V = card K (see Example 7.5 and Example 7.12). Theorem 3.6 (Existence of Basis) Let V be a vector space over a field of scalars K and let E ⊂ V be a free set of V . Then there exists a basis B of V which contains E and such that V = span E ⊕ span(B \ E). Consequently, every non-trivial vector space has a basis.
Proof Consider the family of subsets of V : We shall treat F(E) as a partially ordered set under the inclusion, ⊆. Note that F(E) is a non-empty set, because E ∈ F(E). We observe that every totally ordered subset (chain) C of F(E) is bounded from above by its union C∈C C and also C∈C C ∈ F(E). By Zorn's Lemma F(E) has a maximal element, B . 2. Unlike the case of finite-dimensional vector spaces, in the case of an infinitedimensional vector space V the equality card E = dim V for some free set E of V does not imply that E is a basis of V . Indeed, let B be a basis of V and let E = B \ {v} for some v ∈ B . Then E is a free set with card E = dim V , but E is not a basis for V .
The next result validates the usefulness of the notion of Hamel dimension.
Theorem 3.8 (Isomorphic Spaces) Let V and W be two vector spaces over the same field K such that dim V = dim W . Then V and W are isomorphic. In particular, the mapping σ : V → W , defined by σ( s∈S c s v s ) = s∈S c s w s , is a vector-isomorphism from V to W , where (v s ) s∈S and (w s ) s∈S are bases of V and W , respectively, S is an index set of card S = dim V = dim W and c s ∈ K for all s ∈ S.
Proof The proof is almost identical to the proof of finite-dimensional case and we leave it to the reader. Lemma 3.9 (Subspace Lemma) Let U and V be two vector spaces over the same field K. Then either U and V are isomorphic, or one of the spaces is (can be embedded as) a subspace of the other (see Remark 3.10). Consequently, if U is a vector subspace of V and dim U < dim V , then U is a proper subspace of V .
Proof Let dim U = α and dim V = β . Then exactly one of the following holds: α = β , α < β , α > β , by the axiom of choice in its forth version (Section 2).  The next result is in sharp contrast to its counterpart in the finite-dimensional linear algebra.

Algebraic Dual
We shortly discuss the properties of the algebraic dual V * of an infinite-dimensional vector space V . Both V * and V * * are proper vector space extensions of V . In sharp contrast to the finite-dimensional case however, the vector spaces V , V * and V * * are never isomorphic.
Proof We start from the formula dim V * = (card K) dim V derived in (Jacobson [19], Chapter 9, §5, p.245). Next, assuming ZFC+GCH (Section 2), we show that y x = max{y, 2 x } = max{y, x + } for every two infinite cardinals, x and y. Indeed, if y = ℵ 0 , the formula follows from the fact that ℵ 0 < 2 x . Let y be uncountable. Then y = 2 κ for some infinite cardinal κ by the GCH. Thus y Finally, we let y = card(K) and x = dim V . The second equality in the above formula follows from the first equality since 2 dim V = (dim V) + by the GCH (Section 2).
Notice that dim V * ≥ card K and dim V * ≥ (dim V) + hold trivially. The next equalities follow immediately from the formula in Theorem 4.1.
and V ⊂ B V * , respectively (for an example we refer to # 2 and # 3 in Definition 6.1).

The mapping
For a recent study, from a purely algebraic point of view, of the relationship between the restricted dual V * and the algebraic dual V * of a vector space V with a countable Hamel basis, we refer to the recent article Chirvasitu & Penkov [5] (no relation to solvability of PDE and generalized functions).

Linear Maps and Operators
We present selected results of linear algebra (needed for the rest of the article) which are well-known for finite-dimensional vector spaces, but less-known for infinitedimensional spaces.
Theorem 5.1 (Extension Principle) Let U, V and W be three vector spaces over the same field of scalars, K (the case W = K is not excluded) and let U be a subspace of V . Then every linear map L ∈ L(U, W) can be extended (non-uniquely) to a linear map L ∈ L(V, W).
Then L is an extension of L we are looking for.

Remarks 5.3 (Bracket Notation) We often write T, v instead of T(v)
for the evaluation of T ∈ V * at v ∈ V . In this bracket notation the above definition can be summirized as follows: The above result gives rise to the concept of a regular operator (used in Todorov [37] in the particular case of V = D(R d ) (Example 7.6).
Corollary 5.6 (Solvability) Let V be a vector space and V * be its (algebraic) dual.
Proof An immediate consequence of Theorem 5.4.

Coordinate Isomorphism
We discuss vector spaces K S 0 , which are infinite-dimensional counterpart of the familiar vector spaces K d .
Definition 6.1 (The Space K S 0 ) Let K be a field and S be a non-empty set (wellordered if desired).

We denote by
or simply, e s (t) = δ st for short. We refer to the set {e s : s ∈ S} as the standard (Hamel) basis . The mapping σ : K S 0 → (K S 0 ) * , defined by σ(e s ) = Φ s for all s ∈ S, is the vector space embedding of K S 0 into (K S 0 ) * , which will be written simply as Theorem 6.2 (Properties of K S 0 ) Let K be a field and S be a non-empty set (as above). Then: Proof The part (i) follows immediately from the definition of the spaces K S 0 and Notice that if S is an ordered finite set, the space K S 0 reduces to the familiar K d , where d = card S. Theorem 6.3 (Coordinate Isomorphism) Let K be a field, V be a vector space over K (as before). Let S be a set of card S = dim V and B = {v s : s ∈ S} be a basis of V . Then: , is a vector isomorphism. We call Γ a coordinate isomorphism and f the coordinate function of v relative to the basis B (sometimes the notation f v or even f v,B should be used instead of f ). In the particular case of S = B , we have (ii) follows directly from (ii). Remark 6.4 (Well-Ordered Bases) We often use matrices (including row and column matrices) to visualize the coordinate functions of the vectors and linear operators relative to a particular basis of V . The matrix approach is exceptionally popular in the cases of finite or countable dimensional vector spaces as well as in separable Hilbert spaces. Can we extend the matrix approach to uncountable vector spaces? The answer is yes; we have to invoke the axiom of choice again in the form of its third version (Section 1): Every set, in particular every basis B of V , can be well-ordered. Alternatively, we can well-order the index set S in K S 0 . We call these bases well-ordered bases.

Examples of Infinite-Dimensional Spaces
We present several examples of infinite-dimensional vector spaces and their algebraic duals and demonstrate how to choose an algebraic (Hamel) basis (Theorem 3.6). We shall often rely on the formulas: (Lemma 3.3 and Theorem 4.1) along with the shortcuts presented in Corollary 3.5 and Corollary 4.2. Here is some advice for the order of the calculations: If dim(V) is known (or easy to calculate), we recommend the order of calculations: dim(V) → card(V) → dim(V * ) → card(V * ). If card(V) is known (or easy to calculate), we recommend: Example 7.1 (Hamel Example) Let R|Q denote the Q-vector space of R and (R|Q) * stand for its (algebraic) dual. We have dim(R|Q) = card R = c by Corollary 3.5, since card Q < card R. Also, dim (R|Q) * = max{c + , ℵ 0 } = c + by (2) and card(R|Q) * = c + by Corollary 4.2, since again, card Q < card R. For the original source of this example, we refer to Hamel [12].
Example 7.2 (The Space R|A and its Dual) Let A denote the field of algebraic real numbers. Let R|A denote the A-vector space of R and (R|A) * stand for its dual. As in the previous example, we have dim(R|A) = card R = c and dim (R|Q) * = card(R|A) * = c + since card A = card Q < card R.
by Theorem 6.3. Let us consider the subset       . . ) is a Cauchy sequence, but it is divergent. Indeed, suppose (seeking a contradiction again) that lim n →∞ ||v n − v|| for some v ∈ H. We have v = m k=1 c k w k for some m ∈ N, some c k ∈ C and some w k ∈ B . After replacing, we get lim n →∞ || n k=1 e k k − m k=1 c k w k || = 0, a contradiction, since in (w 1 , · · · , w m , e 1 , e 2 , . . . ) there are not more than finitely many repetitions and the set {w 1 , · · · , w m , e 1 , e 2 , . . . } consisting of mutually orthogonal unit vectors only. Thus, H is non-complete, another contradiction.
• So, we have to choose a non-orthonormal Hamel basis of H, which is desirable to be as much close to an orthonormal as possible. One way to do this is to start from a Hilbert (non-Hamel) orthonormal basis (e 1 , e 2 , . . . ) of H (mentioned already above) and to extend it (non-uniquely) to a Hamel basis B H = {e r : r ∈ R} of H, by Theorem 3.6. Note that the basis B H is non-orthogonal (although it is an extension of an orthonormal Hilbert basis).
• We show now that card H * = dim H * = c + and thus H * ∼ = C P(R) 0 (Section 6). Indeed, we have dim(H * ) = max{(dim V) + , card C} = max{c + , c} = c + by (2). Also, card(H * ) = max{dim H * , card C} = max{c + , c} = c + by (3) Remark 7.10 (An Alternative) Alternatively to the example above, we can define a (new) inner product on H by (v, w) = r∈sp(v) ∩ sp(w)ā r b r , where v = r∈sp(v) a r e r ∈ H and w = s∈sp(w) b s e s ∈ H (Definition 3.4). We observe that sp(e r ) = {r}. Thus, (e r , e s ) = δ rs for all r, s ∈ R. The latter means that B H = {e r : r ∈ R} is an orthonormal Hamel basis of H relative to (·, ·). We observe that (·, ·) coincides with ·, · on H = span{e 1 , e 2 , . . . }, since B H = {e r : r ∈ R} is an extension of {e 1 , e 2 , . . . }. Notice however, that H, (·, ·) is not a Hilbert space by what was explained above; it is non-complete relative to the norm √ (·, ·). Rather, H, (·, ·) is merely an infinite-dimensional inner vector space over C, which admits an orthonormal Hamel basis and which shares with H, ·, · a common inner subspace H . The next several examples are about vector spaces over a field of non-standard complex numbers * C. The reader who is unfamiliar with non-standard analysis (Robinson [33]), might skip these examples and resume the reading from Section 8. Recall that for every infinite cardinal κ there exists a unique (up to a field isomorphism) non-standard extension * C of C in a κ + -saturated ultra-power non-standard model with the set of individuals R (Chang & Keisler [4]; for a presentation we refer also to the Appendix in Lindstrøm [25]). Recall as well that * C is an algebraically closed non-Archimedean field containing C (as a subfield) with card( * C) = κ + . Also, * C = * R(i), where * R is a κ + -saturated real closed non-Archimedean field containing R as a subfield. We should alert the reader that the asterisk in front, * C, has nothing to do with the asterisk after, C * .

Linear Functionals in D * (Ω) as Generalized Functions: The Main Result
We supply D * (Ω) (Example 7.6) with the structure of a sheaf of differential C-vector spaces (and, more generally, a sheaf of differential modules over E(Ω) = C ∞ (Ω)). This structure is inherited from D(Ω) by duality. Our framework is the infinite-dimensional linear algebra presented in Sections 3-5 applied to particular case V = D(Ω) and its dual V * = D * (Ω). As before, Ω stands for a (generic) open subset of R d (Section 2).
We are trying to convince the reader that D * (Ω) deserves to be treated as a space of generalized functions. In what follows the space L loc (Ω) (and more literarily, S Ω [L loc (Ω)] explained below) presents the set of classical functions as apposed to the generalized functions in D * (Ω). The embedding of Schwartz distributions (Vladimirov [39]) in D * (Ω) will be discussed in the next section.
Our approach is a refinement and generalization of distribution theory. That is why, starting from this section, we follow the tradition of distribution theory and use the bracket notation mentioned in (Remark 5.3): We shall write T, ϕ instead of T(ϕ) for the evaluation of T ∈ D * (Ω) at ϕ ∈ D(Ω).

Let
X be an open set of R d such that X ⊆ Ω and T ∈ D * (Ω). We define the restriction T ↾ X ∈ D * (X) of T on X by T ↾X, ϕ = T,φ for all ϕ ∈ D(X), whereφ is the extension of ϕ from X to Ω by zero-values on Ω \ X .
3. Let T n , T ∈ D * (Ω), n ∈ N. We define the weak (pointwise) convergence T n * → T in D * (Ω) by lim n →∞ T n , ϕ = T, ϕ for all ϕ ∈ D(Ω), where lim n →∞ stands for the usual limit in C. 6. Let X and Y be two open sets of R d , θ ∈ Diff(X, Y) be a diffeomorphism from X to Y and J θ : X → Y , J θ = | det ∂θ ∂x |, be the corresponding Jacobian determinant. We define the change of variables θ * :   (ii) The family {D * (Ω)} Ω∈T d is a sheaf of differential modules over E(Ω) relative to the usual topology T d on R d and the restriction ↾ (Kaneko [21], p.16). Consequently, the family {D * (Ω)} Ω∈T d is a sheaf of differential vector spaces over C. (Thus the support, supp(T), is well-defined for every T ∈ D * (Ω)). (iv) The linear partial differential operator P * (x, ∂):
The rest follows immediately from Theorem 5.4 applied for V = D(Ω) and O = P(x, ∂). P roof  Here is an independent proof based on Theorem 3.6 only: Let ran(P(x, ∂)) ⊆ D(Ω) denote the range of P(x, ∂).

Schwartz Distributions within D * (Ω):
Sequential Approach to Distribution Theory We characterize the space of Schwartz distributions D ′ (Ω) (Vladimirov [39]) as a particular subspace of D * (Ω) without involving the usual strong topology on the space of test-functions D(Ω) (Vladimirov [39]). We discuss the similarities and differences between D ′ (Ω) and D * (Ω) and give a short outline of a sequential approach to distribution theory based on our characterization. (ii) D ′ (Ω) is a differential E(Ω)-submodule of D * (Ω). Consequently, D ′ (Ω) is a differential C-vector subspace of D * (Ω).
(iii) The family {D ′ (Ω)} Ω∈T d is a subsheaf of {D * (Ω)} Ω∈T d of differential E(Ω)modules (and C-vector spaces) (Definition 8.1), where T d stands for the usual topology on R d . Consequently, the support supp(T) in D * (Ω) coincides with the usual support of T in distribution theory (Vladimirov [39], §1.5, p.16) for every distribution T .
Proof (i) follows from the fact that the space of Schwartz distributions is sequentially complete under the weak convergence and every distribution can be regularized within D(Ω). For a detailed proof we refer the reader to (Vladimirov [39], §1.4, p.14 and §4.6, p. 80-81). Remark 9.2 (Sequential Approach to Distribution Theory) 1. The formula (4) can be written in the form: where L loc (Ω) N denotes the space of all sequences in L loc (Ω) (Section 2) and the "lim" stands for the usual limit in C.
We borrow the next definition and the following lemma and theorem from (Oberguggenberger [30] p.15).
for all ϕ ∈ D(R d ), whereŤ is the inflection of T (Example 8.2) andŤ * ϕ is the usual convolution in the sense of distribution theory (Vladimirov [39], Ch. 4).

Proof
(i) follows directly from Theorem 8.4, because P * (∂) is a regular operator (Section 11) and D ′ (Ω) ⊂ D * (Ω) (thus δ ∈ D * (Ω)). In both (i) and (ii) we have P * (∂)U = P * (∂)(F ⋆ T) = (P * (∂)F) ⋆ T) = δ ⋆ T = T . The last equality, δ ⋆ T = T , is derived in (ii) and (iii) somewhat differently: (iii) T ⋆ δ, ϕ = T,δ ⋆ ϕ = T, δ ⋆ ϕ = T, ϕ , for all ϕ ∈ D(R d ) (or even for all ϕ ∈ E(R d )).  [39] (see also Remark 9.2 in this paper). However, there are also essential differences; here are some of them: (1) The discontinuous (relative to the strong topology on D(Ω)) linear functionals T ∈ D * (Ω) \ D ′ (Ω) cannot be approached by a sequence of classical functions in the sense that there is no sequence (f n ) in L loc (Ω) such that f n * → T (Definition 8.1). In a sense the spaces L loc (Ω) (more precisely, S[L loc (Ω)]), D ′ (Ω) and D * (Ω) resemble Q, R and C, respectively: Every real number is the limit of some (fundamental) sequence in Q, but the complex numbers of the for form a + ib, b = 0, can not be approximated by sequences in Q (we are unaware of subspace of D * (Ω) which plays the role of Q(i) in the above anlalogy).
Recall that the structural theorem states that for every Schwartz distribution T ∈ D ′ (Ω) and for every open set X of R d , such that X ⊂⊂ Ω, there exist a classical function f ∈ L ∞ (X) and a multi-index α ∈ N d 0 such that T = ∂ α f in D ′ (X). This theorem fails in D * (Ω).
(3) The convolution (Definition 9.4) fails to regularize T ∈ D * (R d ) \ D ′ (R d ) in the sense that T ⋆ ϕ ∈ E(R d ) does not necessarily hold for all ϕ ∈ D(R d ).
(5) From the above list it seems that the space D ′ (Ω) is superior over D * (Ω) at least from the point of view of partial differential operator theory. As we shall see in the next sections however, the regular operators P * (x, ∂) are (always) surjective on D * (Ω), but their restrictions on D ′ (Ω) are often not. That means that the partial differential equations of the form P * (x, ∂)U = T are always solvable for U in D * (Ω), but not necessarily solvable in D ′ (Ω). The latter property of D * (Ω) is the main reason why we believe that the space D * (Ω) -rather than D ′ (Ω) -should be considered as the natural framework of partial differential equations, especially the linear ones with smooth coefficients.

Three Invariant Subspaces
In order to prepare better for the discussion in the next section, we select three important subspaces of D * (Ω), which are invariant under the linear partial differential operators with C ∞ -coefficients. The restrictions P * (x, ∂) ↾ D(Ω), P * (x, ∂) ↾ E(Ω) and P * (x, ∂) ↾ D ′ (Ω) of the regular operators P * (x, ∂) in our examples below are relatively well-studied in the classical theory of linear partial differential operators (Hörmander [15]- [17]). With few exceptions, these restrictions are non-surjective. Thus we arrived at the main point of our approach -to extend non-surjective operators from D ′ (Ω) to surjective operators on D * (Ω) and thus to guarantee the existence of solutions for the corresponding partial differential equations in the framework of D * (Ω).
Standardizations of non-standard results are fascinating and often dramatic events in mathematics: • The most famous example of standardization is certainly the creation of the modern calculus which standardizes the old Leibniz-Newton-Euler Infinitesimal Calculus. Getting rid of infinitesimals and replacing them with limits can be described as nothing less than a real drama or even revolution in mathematics (Hall & Todorov [10]). The drama took several decades and it resulted -as side products -in the rigorous theory of real numbers used today, set theory and mathematical logic. For those interested in the history of infinitesimals in the context of the Reformation, Counter-Reformation and English Civil War, we refer to the excellent book by Alexander [1].  [26] in the framework of a more advanced and general operator theory.