Abstract Methods in Partial Differential Equations

By Robert W. Carroll

This self-contained text is directed to graduate students with some previous exposure to classical partial differential equations. Readers can attain a quick familiarity with various abstract points of view in partial differential equations, allowing them to read the literature and begin thesis work. The author's detailed presentation requires no prior knowledge of many mathematical subjects and illustrates the methods' applicability to the solution of interesting differential problems.
The treatment emphasizes existence-uniqueness theory as a topic in functional analysis and examines abstract evolution equations and ordinary differential equations with operator coefficients. A concluding chapter on global analysis develops some basic geometrical ideas essential to index theory, overdetermined systems, and related areas. In addition to exercises for self-study, the text features a thorough bibliography. Appendixes cover topology and fixed-point theory in addition to Banach algebras, analytic functional calculus, fractional powers of operators, and interpolation theory.

• Language: English
• Publisher: Dover Publications
• Release date: May 27, 2013
• ISBN: 9780486263281

Book preview

CARROLL

CHAPTER 1

Functional Analysis and Distributions

1. INTRODUCTION

This chapter consists mainly of a concise introduction to the theory of distributions following L. Schwartz. The subject is viewed as a topic in the theory of locally convex topological vector spaces (LCS) and is used simultaneously as a model and vehicle to develop some locally convex theory very rapidly. The remainder of the book will not require much more about distributions than a good understanding of what they are; however, there are several reasons beyond that for dealing with the subject somewhat more thoroughly here despite a number of excellent recent textbook-style expositions (see references below). First, much of the functional analysis will be needed later as well as, for example, the notion of tensor product. Second, distributions are an essential ingredient for modern work in partial differential equations, and we feel it is appropriate and necessary to discuss them seriously in a book about abstract methods in PDE. In addition, distributions represent much more that a technical device to solve PDE and we would like to illustrate and emphasize to the uninitiated some of the beauty of the subject as a topic in functional analysis. The basic reference is [S 1], and we cite also [Fr 1; Ge 1, 2; H 1; Gx 1; Ho 1; Tr 3; Ncb 1; Seb 1, 2]. For general topology see [Ke 1] or [B 1], and for topological vector spaces see, for example, [B 2, 3; Ko 1; G 1; Tr 4], and the references on distributions above. We will not treat extensions of distributions related to boundary values of analytic functions (such as ultradistributions, hyperfunctions, etc.) but refer for this to [Seb 3, 4, 5; Tm 1, 2; Sat 1; Bgl 1; Ko 2; Ru 1, 2; Mt 1, 2; Eh 3; Har 1; Ks 4]. Similarly, we will not discuss Mikusinski operators (see, e.g., [Mki 1]).

2. LOCALLY CONVEX SPACES

The theory of differential operators can be developed in terms of unbounded (not continuous) operators in Hilbert and Banach spaces and simultaneously in terms of continuous operators in somewhat more exotic spaces. These points of view are not really distinct, of course, and we should like to emphasize the interaction between them rather than treat them as separate doctrines. In fact, we shall work primarily in Hilbert spaces in much of this book but distribution techniques will frequently intervene.

′. The setting is very productive mathematically and certain basic types of theorems about differential operators have an if, and only if, version in suitable distribution spaces (cf. [H 1; Tr 5; Bj 1]). This is entirely natural, of course; one starts with convergence defined in terms of derivatives and ends up with good theorems about differentiation and differential equations. We shall tailor our choice of material, beyond the fundamentals, to the needs of the rest of the book, and thus many important things will be omitted, or simply remarked upon, when they fall outside the development we adopt. The arguments will generally be phrased in a way which generalizes to other locally convex situations.

We begin by collecting a few standard facts and definitions, and since topological vector spaces are so basic in partial differential equations, the details will be spelled out (quickly).

Definition 2.1

A (left) vector space F over a field K is a collection of elements satisfying

(a) F is an abelian group under +;

, xx : K × F → F (x + yx y)x x xx)x;

(c) The multiplicative identity 1 of K acts as an identity in F in the sense that 1x = x for all x ∈ F.

Definition 2.2

A topological vector space (TVS—we suppress the word left from now on) is a vector space F endowed with a topology satisfying

, xx : K × F → F is continuous;

(b) (x, y) → x + y : F × F → F is continuous;

(c) x → − x : F → F is continuous.

In addition, we shall assume that all TVS are Hausdorff spaces unless otherwise stated and K will always be C or R (in a Hausdorff space for any x, y, x ≠ y, there are open sets—see below—U, V, x ∈ U, y ∈ V such that U ∩ V = ∅ where ∅ is the empty set).

Definition 2.3

A seminorm in a vector space F over K (= C or R) is a function p : F → R satisfying

(a) p(x + y) ≤ p(x) + p(y);

(b) px|p(x).

If, in addition, p(x) = 0 implies x = 0, then p is called a norm.

We recall now that a topology on a collection of objects F is defined by the prescription of what are to be open sets, where, to qualify as a family of open sets, a family O of subsets of F must satisfy

Then one defines a neighborhood (nbh) of a point to be any set containing an open set containing the point and we denote by N(x) the family of nbhs of x. It is elementary to verify that N(x) satisfies the following four rules.

Definition 2.4

Neighborhood axioms:

(a) Every set in F containing a set in N(x) itself belongs to N(x).

(b) Ni ∈ N(x(finite intersection).

(c) x ∈ N if N ∈ N(x).

(d) If V ∈ N(x), there exists W ∈ N(x) such that for all y ∈ W, V ∈ N(y).

Here (a), (b), (c) are obvious and (d) follows upon taking W to be an open set containing x and contained in V and observing that a set is open if and only if it is a nbh of each of its points. It is a standard theorem of general topology (see [B 1]) that if each x ∈ F has associated to it a family N(x) satisfying (a) to (d) of definition 2.4, then there is a unique topology on F such that N(x) is precisely the family of nbhs of x for this topology. In fact, one can describe this topology by picking as open sets those sets 0 such that for each x ∈ 0 one has 0 ∈ N(x) (Exercise 1). We shall call a family B(x) ⊂ N(x) a base of N(x) or a fundamental system of neighborhoods (fsn) at x if every N ∈ N(x) contains a set B ∈ B(x). Then N(x) is recovered as the family of sets containing a set in B(x). On the other hand, if an arbitrary family B(x) is prescribed for each x, then it qualifies as a fsn at x [with N(x) then defined as the family of sets containing a set in B(x)], if for example, each B ∈ B(x) contains x, B1, B2 ∈ B(x) implies B1 ∩ B2 ⊃ B3 ∈ B(x), and axiom (d) of definition 2.4 holds for B(x) with conclusion V ⊃ a set in B(y); N(x) then satisfies (a) to (d) of definition 2.4. These conditions on B(x) will be frequently used to identify some particular B(x) as a fsn at x and will be infrequently called the standard conditions for a fsn.

Now, to construct the distribution spaces, first let Km ⊂ Km + 1 be an increasing sequence of compact sets in Rn m be the vector space of all infinitely differentiable (i.e., C ) complex-valued functions with support in Km. The support of a function is defined to be the closure of the set of points x where f(xm l l and all its derivatives Dp l [see (2.4)] tend to zero uniformly on Km (uniformity in p is not required). This easy concept will now be formalized in a way leading to much future economy of thought. We define seminorms Np m by

where p = (p1,…,pn) and

The factor 1/i is introduced now so as to have a uniform notation later, since it is convenient to define Dk

for |p| ≤ s. We take now as a fsn Bm all sets of the form V(Km, 1/n, s) for n = 1, 2, … and s = 0, 1,… and then specify B(x) = x + B(0) for any x m. It is easily checked that B(x) satisfies the standard conditions mentioned above for a fsn and consequently that N(x) = x + Nm containing a set in B(xm with these nbhs is a vector space and has a topology determined by the N(xm is a TVS it remains only to check the continuity conditions of definition 2.2. To do this we recall first

Definition 2.5

If F and G are topological spaces, then a map f : F → G is continuous at x if for any W ∈ N(f(x)) there is a V ∈ N(x) such that f(V) ⊂ W. In view of the nbh structure for TVS [that is, N(x) = x + N(0)], one need only check continuity at 0 for linear maps between TVS.

It is easily checked that this is equivalent to the requirement that f− 1(U) be open for any open U. Now, the continuity properties of definition 2.2 follow immediately from the definition N(x) = x + NV(Km, s) = V(Km, s, then W + W ⊂ V [thus (x + W) + (y + W) ⊂ x + y + Vm is a TVS and, in fact, an LCS, where we define

Definition 2.6

An LCS (locally convex topological vector space) is a TVS whose topology is defined by a family of seminorms p . Thus if V ) = {x ∈ F; p (x}, then a fsn B(0) is taken to be the family of finite intersections of the V ), while if, for any x ∈ F, there is a p with p (x) ≠ 0, then F is Hausdorff. A morphism E → F of TVS (or LCS) is a continuous linear map f : E → F; f is a homomorphism if it is also open [i.e., f(0) is open for 0 open] and an isomorphism if it is 1 − 1 and bicontinuous.

m we picked out a subclass {V(Km, 1/n, s)} of the family of all finite intersections of the Vp(Km), but it is easy to see that one gets the same family of nbhs N(0) and hence N(x) (verify this). It is evident that the seminorms p now become continuous functions F V ) = V )].

Definition 2.7

A metric space is a set F with a distance function or metric d : F × F → R satisfying

(a) d(x, y) ≥ 0;

(b) d(x, y) = 0 if and only if x = y;

(c) d(x, y) = d(y, x);

(d) d(x, z) ≤ d(x, y) + d(y, z).

A TVS is called metrizable if there is at least one metric defined on it such that the sets Vn = {y ∈ F; d(x, y) ≤ 1/n} form a fsn at x.

m is metrizable is to produce an admissible metric. To do this we observe that V(Km, s) is associated with the seminorm ns = sup Np for |p| ≤ s [i.e., V(Km, sm; ns}]. Consider the countable fsn (n and s varying) V(Km, 1/n, s) with associated seminorms ns and define the family of seminorms q1 = n1, q2 = sup(n1, n2), …. Then q1 ≤ qand sets of the form Ws(1/nm; qs) ≤ 1/n} form a fsn giving rise to the same N(0). Define

and check that d(x, y) = p(x − ym m . We recall next

Definition 2.8

A net in F is a pair (S, ≥), where S is a function with values in F and ≥ directs the domain A of S ≥ y ∈ A ∈ A ). A net (S, ≥ ), denoted also by S or {S }, is eventually in a set B implies S ∈ B. A net converges to x if and only if it is eventually in every nbh of x.

of B consists of all points x such that every nbh of x intersects B, we see that if x , then there is a net S in B converging to x. Indeed, direct the nbhs U of x by inclusion and pick a point SU in each U ∩ B. In a metric TVS F the topology is defined by a countable fsn at 0 formed of sets Vn F; d0) ≤ 1/n}. Hence x ⊂ F if and only if there is a sequence in B converging to x. Thus, in general, B is closed if and only if no net in B (or sequence if F is metric) converges to a point of F − B. This means that closed sets are completely described by convergence properties, and since closed sets are the complements of open sets we can completely describe the topology in terms of convergence (see [Ke 1; B 1] for further discussion).

Definition 2.9

A net S in a TVS F is a Cauchy net if, given any V ∈ Nsuch that S − S ∈ V . A space F is complete if every Cauchy net converges; if F is metric (or metrizable) it is complete if every Cauchy sequence converges. A complete metrizable LCS is called a Frechet space and a complete normed space is called a Banach space. A complete TVS which possesses a Banach structure but which is being considered only as a TVS is called a Banachable space (i.e., one forgets the norm and considers only the topology).

Theorem 2.10

m is a Frechet space (cf. exercise 3).

be the vector space of all C functions on Rn m for each Km and we denote by im m the finest locally convex topology such that all the im are continuous. Here finest means strongest in the sense of having the largest collection of open sets (or nbhs). That such a finest topology is well defined follows from an elementary filter argument (see [B 1, 2]) but we shall avoid a digression here by simply constructing the topology and then showing it is finer than any (other) locally convex topology having the im all continuous. We recall first that a set B is disced if dx ∈ B whenever x ∈ B and |d| ≤ 1. Further, if F is a TVS over R, then B is convex if x, y ∈ B x )y ∈ B < 1. If F is a TVS over C, let F0 be the same space considered as a TVS over R (multiplication by i, then, is treated as an automorphism of F0 and not as a dialation). Then B ⊂ F is convex if it is convex in F0. Now if Vm ∈ Nm as Vm runs over Nm and consider it as a fsn at 0 for a topology T , we see that im is indeed continuous. On the other hand, if im is continuous for a topology Tfor all m and V is a (closed) convex disced nbh of 0 for T′, then V and V m ∈ Nm[note that the continuity of a function f : F → G means f− 1(W) ∈ N(x) for any W ∈ N(f(x)), where f− 1 means the complete inverse image]. Hence V ∈ N(0) for T. Observe next that if T′ is a locally convex topology, then it has a fsn composed of (closed) convex disced sets. To see this, recall definition 2.6 and note that the V ), and their finite intersections, are convex (in F0), closed, and disced. Hence we have shown T ⊃ T′ for any locally convex topology Tfor which im is continuous, since any T′ nbh U is also a T nbh (T ⊃ T′ means T is finer than T′ or T′ is coarser than T). It remains to show that in fact T is another sequence of compact sets exhausting Rn. Hence we shall always think of some fixed sequence Km .

Definition 3.1

with the topology T m .

Remark 3.2. is locally convex for the limited use to be made of distributions in this book, but it is easy enough to show that it is. We give a brief discussion of this here. First recall that a TVS F is usually said to be locally convex if N(x) in F, xx one knows that x B 0 (B is said to be absorbing). Define gauge B as the function

Then (cf. [B 2]) p = {x ; p(xis locally convex by our definition. In this connection we recall also a standard fact from general topology (cf. [B 1, 2, 4]). A space is regular (by definition) if the set of closed nbhs of an arbitrary point x is a fsn at x, Vm = V(Km, s), then 3Wm = Wm + Wm + Wm ⊂ Vm m is not a metrizable space, as is suggested (but not proved) by the use of a noncountable family of seminorms in defining the topology. A proof can be based on the Baire category theorem, for example, and we refer the matter to the exercises (Exercise 5).

now permits us to reduce many arguments to metric spaces and hence to sequential convergence.

Theorem 33

A linear map f → F, F a LCS, is continuous if and only if the restriction fm = fm m → F is continuous for each m.

Proof

A linear f → F is continuous if and only if f− 1(V) ∈ Nfor V in a fsn at 0 in F. Thus take V convex and disced, in which case f− 1(V) is convex and disced. Then f− 1(V) ∈ Nif and only if f− 1(Vm ∈ Nm. This is equivalent to saying that fm is continuous. QED

We remark that other equivalent descriptions of the topology T are possible (see [S 1; H 1]). We give here the analytical description furnished by Schwartz [S 1]. Thus let {Ω} = {Ω0 = ∅, Ω1, …} be a sequence of open sets, Ωn − 1 ⊂ Ωn, such that any compact K ⊂ Rn is eventually contained in the Ωnn → 0, and {m} = {m0, m1,…} a sequence of positive integers increasing toward + ∞. Let V({msuch that for x Ωn, |Dp (xn if |p| ≤ mn}, {m} vary, the V({m}, {Ω}) form a fsn for the topology T (see Exercise 6). One should also note explicitly that the topology induced by T m m (see below).

′ AND EXAMPLES

We now define the Schwartz distributions.

Definition 4.1

If F is a TVS over C the (topological) dual of F is the vector space of all continuous linear maps f′: F → C. The action of f′ on f will be written 〈f′, f〉 = 〈f, f′〉 = f′ (ff′, f〈f′, f〉 and 〈f′ + g′, f〉 = 〈f′, f〉 + 〈g′, fis called the space of Schwartz distributions in Rn.

Observe that the dual space has not been given a topology yet; several topologies are interesting and we shall discuss this later. Now look at Dk = (1/ixk m m . But Dk m m m m topology, Tm say. To see this let T′m m. Since im m is continuous, we know T′m ⊂ Tm. Conversely, pick some Vm = V(Km, s) and define, for example, Vp = V(Kp, s) for all p (note Vp ⊂ Vp. Consequently, Tm ⊂ T′m. But Dk m m is obviously continuous and we have proved

Theorem 4.2

Dk is a continuous linear map.

We consider now the linear map

for T ′ given. Since Dk is continuous and T → C or a distribution S ′. Thus we write

and define S = DkT. This formula reduces to integration by parts when T and S .

Definition 4.3

′ is defined for any T ′ by the formula (4.2) and we write S = DkT. Thus Dk ′.

→ C and we refer the proof to the exercises).

Example 4.4. Any locally summable function f defines a distribution by the rule

Example 4.5. is defined by

to obtain

Example 4.6. The Heaviside function Y in R¹ is defined as Y = 0 for x ≤ 0 and Y = 1 for x ≥ 0. Then (′ ≡ d/dx)

and we write Y.

Example 4.7. n n/2Γ(n/2) be the surface area of the unit ball in Rn; then rnn is the surface area of a ball of radius r centered at the origin. Define two mean value measures by

where d denotes the surface-area element. Then for r x(r) and Ax(r′ in Rn′-valued functions of r and for now we note only the following formal calculations (using Green’s theorem):

It will be seen in section 11 that this formula can be rigorously interpreted to mean

Similarly,

which leads to

Such equations are important in the theory of the Euler-Poisson-Darboux (EPD) equation (see, e.g., [C 1, 2, 3; W 1, 2] and references later to singular problems).

is locally convex. Although such recourse to local convexity and Hahn-Banach has been deliberately avoided in our development, in the interest of simplicity, we shall state the relevant facts here since Hahn-Banach will be needed later (see theorem 10.8 for a geometrical version of the Hahn-Banach theorem).

Theorem 4.8 (Hahn-Banach)

Let p(·) be a continuous seminorm in a (complex) LCS F and suppose given a linear form l defined on a closed linear subspace G ⊂ F such that |l(z)| ≤ p(z) for z ∈ G. Then l may be extended to a continuous linear form u on all F with |u(x)| ≤ p(x) for x ∈ F.

Proof

We prove the theorem for real spaces in remarking that the complex case can then be obtained by an elementary calculation (cf. the discussion before theorem 10.8). For l defined as above on any closed linear sub-space G ⊂ F and xG, we will show that l can be extended to the linear space G′ spanned by x0 and G while preserving the properties indicated. Then by Zorn’s lemma or transfinite induction one can extend to all of F. Thus for any z, z′ ∈ G, note that

This holds for any z, z′ ∈ G and hence

be a number between these extremes and define l on G′ by lx0 + z+ l(z) for z ∈ G. To show that l(y) ≤ p(y) on G> 0; then since z∈ G we have

= − s, s > 0; then z= − z/s ∈ G and

Since p is now continuous at 0, it follows that p(xfor x in some neighborhood V of 0 in F. But we can always choose a fsn which is symmetric (i.e., x ∈ V implies − x ∈ V) in an LCS. Then suppose the extension of f to have been carried out to all F with l(x) ≤ p(x) on F. From l(x) ≤ p(xfollows also now − l(x) = l (− x) ≤ p (− x. Therefore, |l(xfor x ∈ V and l is continuous. Also this shows that |l(x)| ≤ p(x) since − l(x) = l(− x) ≤ p(− x) = p(x). QED

The same proof applies in any TVS since symmetric nbhs as above can be found, but we omit the details because we only deal with LCS in this book. As a corollary we have

Corollary 4.9

For any continuous seminorm p on a LCS F there is a u ∈ F′ with |u(z)| ≤ p(x) and u(x0) = p(x0), x0 arbitrary, can be prescribed in advance.

Proof

Define l on {x0} by lxp(x0) and extend l; here {xx0. QED

Thus, in particular, in a LCS F, where continuous seminorms exist, there are automatically nontrivial elements of F′.

′ (R¹)

′ on R¹ (following Schwartz [S 1]) whose vector-valued versions (cf., for example, theorem 11.11) are needed later.

Theorem 5.1

Given S ′ on R¹ there are an infinite number of primitives T ′ (i.e., T′ = S) and any two differ by a constant.

Proof

The proof given in [S 1] is instructive and we follow it. In order that T′ = S for T ∈ D′ it is necessary and sufficient that 〈T〉 = 〈T′〉 = − 〈S〉 for all x of the form x forms a (complex) hyperplane H (actually a subspace, since 0 ∈ H). Recall that a hyperplane P ⊂ F, F a complex TVS, is defined as the translate of a subspace H0 ⊂ F determined by H0 = {x ∈ F; f(x) = 0}, where f is a complex linear form on F; thus P = {x : f(x}. Alternatively, H0 can be characterized as a subspace with dimC F/H0 = 1 (see [B 5; La 1] for the algebra, which we assume known). If f is continuous, then H0 = HmH . Thus given S ′, we define a linear form T by

′ and 〈Tk mk k k 0 → 0 in H q, where Kq ⊃ Km k qq and consequently 〈Sk〉 = − 〈Tk〉 → 0, since S ′. Hence we have shown that 〈Tkk m for any m and therefore T ′. Evidently, T′ = S and any primitive is of the form (5.1). Finally, if T′1 = T′2 = S, then form (5.1),

where c = 〈T1 − T0〉. Hence T1 − T2 = c. QED

We assume standard facts about Lebesgue integration and Lp spaces here [a unified discussion of this will be given in f ∈ Lp ′ (with abuse of notation); the embedding really involves looking at f as a measure f dx in this situation. Of course, one must be careful to avoid incorrect lifting procedures, for example, Lp p (p ≠ ∞). We will spell out arguments where abuse of notation might lead to some confusion. The usual derivative of f will be denoted by [f′], and f′.

Theorem 5.2

Let f be an absolutely continuous function with [f′] = g a.e. and . Then f′ = g. Conversely, if f ′ and , then f is an absolutely continuous function and [f′] = g.

Proof

If [f′] = g this means f′ = g ′. Conversely, if f′ = g and f for some a. Then, as above, h′ = g ′ and, by theorem 5.1, f = h + c ′. Hence f can be identified with an absolutely continuous function as indicated. QED

′ is used as the identification space and f ′ is shown to be a function h + c ′ (i.e., as a distribution). It is not necessary to assume in advance that f is a function. We mention explicitly that distributions T do not have values T(x) at a point x but one can define a notion of fixing variables as in [Lj 2] (see [C 2; Shr 1] for some further applications of this).

′ AND THE FOURIER TRANSFORM

Next we define some other spaces and some operations.

Definition 6.1

be the space of C functions on Rn (no restriction on supports) with topology defined by the following family of nbhs as a fsn. Let Km ⊂ Km and define Np), Vp(Km), V(Km, s. Then take the sets

as a fsn at 0 as m, n, and s mis a Frechet space.

Definition 6.2

be the space of C functions on Rn, decreasing as |x| → ∞, along with all their derivatives, faster than any power of 1/|x|. Thus if P(x) is any polynomial and Q(D) any polynomial in the symbols DK, then P(x)Q(Dis a continuous bounded function on Rn we take sets of the form

Thus one is dealing with seminorms Nm, k = sup Np, k for |p| ≤ m, where Np, k) = sup(1 + |x|²)k|Dp | for x ∈ Rnis a Frechet space.

′.

Definition 6.3

If T ′, then T is said to be zero in an open set Ω ⊂ Rn if 〈T(Ω) is the space of C functions with compact support lying in Ω. The union of all open Ω where T = 0 is open and its complement is defined as the support of T (written supp T or T). Thus supp T is the smallest closed set outside of which T = 0 and x ∈ supp T if T ≠ 0 in any open nbh of x.

Theorem 6.4

′ is the space of distributions T ′ with compact support.

Proof

Let T ′ with supp T k . Pick a C = 1 in some nbh Km of supp T = 0 outside Km + 1 ⊃ Km (we will show in section 9 how to construct such functions explicitly); note that Km a nbh of supp T means there is an open set 0 with Km ⊃ 0 ⊃ supp Tk m + 1. Hence 〈TkT′ duality) to be 〈Tis another such function, then

supp Tin the nbh of supp Tk Tk → 0 and hence that T ′.

→ LL k mk and hence L k) → 0. Hence there exists T ′ such that L) = 〈T. Moreover, supp T k k(x) = 0 for |x| ≤ k and 〈Tkk and hence Lk) = 〈Tk〉 would have to vanish as k → ∞. Thus T j j j on a compact nbh of supp Tj j = 1 on Bj = {x ∈ Rn; |x| ≤ jj . Start counting for j > j0, where supp T ⊂ Bjo j j . Then Lj) → L) and 〈TjTj) = 〈TjT= Land thus L = T ′. QED

In general we shall write 〈 , 〉 for any duality when no confusion can arise.

Definition 6.5

is defined by

. The inverse transform is

To see that this makes sense note that

where (− x)p = (− x1)p (− xn)pn. Consequently,

where P and Q are polynomials. In view of the rapid decrease of the functions P(DX){Q (− x}, the integrals are easily bounded and in fact one can immediately show that F is continuous (Exercise 9). The classical inversion formula will be assumed (cf. [Ti 1]) and hence we can state (applying the same arguments to F)

Theorem 6.6

F is a continuous linear map onto with a continuous inverse (i.e., F is an isomorphism of LCS).

(see [Ti 1]):

Let now u ′ duality)

Since F →