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Abstract Methods in Partial Differential Equations
By ROBERT W. CARROLL
Dover Publications, Inc.Copyright © 1997 Robert W. Carroll
All rights reserved.
Functional Analysis and Distributions
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.
Roughly speaking, distribution theory, or the theory of generalized functions, is all about differentiation. One constructs a space D of infinitely (i.e., indefinitely) differentiable functions with compact support (definition below) and places on it a very fine topology (i.e., lots of open sets), defined in terms of convergence of a function and all its derivatives (in a suitable sense). Then the dual D' of all continuous linear forms (or functionals) on D is a very big space and differentiation can be defined in D' by duality as a continuous linear map D' -> D'. This enables one to differentiate a large class of objects, including the delta "functions" of physics, as often as desired, and many of the formal calculations of physics can be made rigorous in this framework. Fourier analysis also has a natural setting in a big space, L' [subset] D'. 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).
A (left) vector space F over a field K is a collection of elements satisfying
(a) F is an abelian group under +;
(b) There is a map (α, x) -> αx : K × F -> F such that α(x + y) = αx + αy, (α + β)x = αx + βx, and α(x) = (αβ)x;
(c) The multiplicative identity 1 of K acts as an identity in F in the sense that 1x = x for all x [member of] F.
A topological vector space (TVS—we suppress the word "left" from now on) is a vector space F endowed with a topology satisfying
(a) (α, x) ->x : 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 [member of] U, y [member of] V such that U [intersection] V = [empty set] where [empty set] is the empty set).
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) p(αx) = |α|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
(2.1) [union]0α [member of] O any index set, [empty set] [member of] O
(2.2) [union]0n [member of] O finite intersection F [member of] O
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.
(a) Every set in F containing a set in N(x) itself belongs to N(x).
(b) Ni [member of] N(x) implies [intersection] Ni [member of] N(x) (finite intersection).
(c) x [member of] N if N [member of] N(x).
(d) If V [member of] N(x), there exists W [member of] N(x) such that for all y [member of] W, V [member of] 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 [member of] 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 [member of] 0 one has 0 [member of] N(x) (Exercise 1). We shall call a family B(x) [subset] N(x) a base of N(x) or a fundamental system of neighborhoods (fsn) at x if every N [member of] N(x) contains a set B [member of] 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 [member of] B(x) contains x, B1, B2 [member of] B(x) implies B1 [intersection] B2 [contains] B3 [member of] B(x), and axiom (d) of definition 2.4 holds for B(x) with conclusion V [contains] 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 [member of] Km + 1 be an increasing sequence of compact sets in Rn with [union] Km = Rn (a set is compact if every open covering has a finite subcovering). Let Dm 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(x) ≠ 0. We want to define a topology in Dm so that a sequence φl -> 0 whenever φl and all its derivatives Dpφl [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 in Dm by
(2.3) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]
where p = (p1, ..., pn) and
(2.4) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]
The factor 1/i is introduced now so as to have a uniform notation later, since it is convenient to define Dk this way when dealing with the Fourier transform. Define further the sets (|p| = [summation]pi)
(2.5) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]
(2.6) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]
Then evidently V(Km, ε, s) = [union] Vp(Km, ε) for |p| ≤ s. We take now as a fsn B(0) in Dm 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 [member of] Dm. It is easily checked that B(x) satisfies the "standard" conditions mentioned above for a fsn and consequently that N(x) = x + N(0), the family of sets in Dm containing a set in B(x), satisfies the conditions of definition 2.4. Then Dm with these nbhs is a vector space and has a topology determined by the N(x). To verify that Dm is a TVS it remains only to check the continuity conditions of definition 2.2. To do this we recall first
If F and G are topological spaces, then a map f : F -> G is continuous at x if for any W [member of] N(f(x)) there is a V [member of] N(x) such that f(V) [subset] 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 + N(0). Indeed (a) and (c) are obvious [note that αV(Km, ε, s) = V(Km, αε, s)] and (b) follows directly from the observation that if W = 1/2V(Km, ε, s) = 1/2V, then W + W [subset] V [thus (x + W) + (y + W) [subset] x + y + V]. Therefore, Dm is a TVS and, in fact, an LCS, where we define
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