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Examples of Function Spaces 1. Non-Banach Limits C K(R)

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Examples of Function Spaces 1. Non-Banach Limits C K(R) (February 11, 2017) Examples of function spaces Paul Garrett [email protected] http:=/www.math.umn.edu/egarrett/ [This document is http://www.math.umn.edu/~garrett/m/fun/notes 2016-17/examples.pdf] 1. Non-Banach limits Ck(R), C1(R) of Banach spaces Ck[a; b] k k 2. Banach completion Co (R) of Cc (R) 3. Rapid-decay functions, Schwartz functions 1 4. Non-Fr´echet colimit Cc (R) of Fr´echet spaces 5. LF-spaces of moderate-growth functions 6. Strong operator topology 7. Generalized functions (distributions) on R 8. Tempered distributions and Fourier transforms on R 9. Test functions and Paley-Wiener spaces We review natural topological vectorspaces of functions on relatively simple geometric objects, such as R or the circle T. In all cases, we specify a natural topology, in which differentiation or other natural operators are continuous, and so that the space is suitably complete. Many familiar and useful spaces of continuous or differentiable functions, such as Ck[a; b], have natural metric structures, and are complete. Often, the metric d(; ) comes from a norm j · j, on the functions, giving Banach spaces. Other natural function spaces, such as C1[a; b], Co(R), are not Banach, but still do have a metric topology and are complete: these are Fr´echetspaces, appearing as (projective) limits of Banach spaces, as below. These lack some of the conveniences of Banach spaces, but their expressions as limits of Banach spaces is often sufficient. o Other important spaces, such as compactly-supported continuous functions Cc (R) on R, or compactly- 1 supported smooth functions (test functions) D(R) = Cc (R) on R, are not metrizable so as to be complete. Nevertheless, some are expressible as colimits (sometimes called inductive limits) of Banach or Fr´echet spaces, and such descriptions suffice for many applications. An LF-space is a countable ascending union of Fr´echet spaces with each Fr´echet subspace closed in the next. These are strict colimits or strict inductive limits of Fr´echet spaces. These are generally not complete in the strongest sense, but, nevertheless, as demonstrated earlier, are quasi-complete, and this suffices for applications. 1. Non-Banach limits Ck(R), C1(R) of Banach spaces Ck[a; b] For a non-compact topological space such as R, the space Co(R) of continuous functions is not a Banach space with sup norm, because the sup of the absolute value of a continuous function may be +1. o But, C (R) has a Fr´echet-space structure: express R as a countable union of compact subsets Kn = [−n; n]. o o o Despite the likely non-injectivity of the map C (R) ! C (Ki), giving C (R) the (projective) limit topology o o o limi C (Ki) is reasonable: certainly the restriction map C (R) ! C (Ki) should be continuous, as should o o all the restrictions C (Ki) ! C (Ki−1), whether or not these are surjective. o o The argument in favor of giving C (R) the limit topology is that a compatible family of maps fi : Z ! C (Ki) gives compatible fragments of functions F on R. That is, for z 2 Z, given x 2 R take Ki such that x is in the interior of Ki. Then for all j ≥ i the function x ! fj(z)(x) is continuous near x, and the compatibility assures that all these functions are the same. 1 Paul Garrett: Examples of function spaces (February 11, 2017) That is, the compatibility of these fragments is exactly the assertion that they fit together to make a function o x ! Fz(x) on the whole space X. Since continuity is a local property, x ! Fz(x) is in C (X). Further, there is just one way to piece the fragments together. Thus, diagrammatically, o * o ' o C (R) ::: / C (K2) / C (K1) 4 c f v; f j j 2 v 1 j j z!F v j j z v j j Z jv j o o k k Thus, C (X) = limn C (Kn) is a Fr´echet space. Similarly, C (R) = limn C (Kn) is a Fr´echet space. o o o o [1.1] Remark: The question of whether the restriction maps C (Kn) ! C (Kn−1) or C (R) ! C (Kn) are surjective need not be addressed. Unsurprisingly, we have d k k−1 [1.2] Theorem: dx : C (R) ! C (R) is continuous. d 1 1 Proof: The argument is structurally similar to the argument for dx : C [a; b] ! C [a; b]. The d k k−1 differentiations dx : C (Kn) ! C (Kn) are a compatible family, fitting into a commutative diagram * * k−1 k−1 k−1 C (R) ::: / C (Kn+1) / C (Kn) / ::: O O d d dx dx k k k C (R) ::: / C (Kn+1) / C (Kn) / ::: 5 4 Composing the projections with d=dx gives (dashed) induced maps from Ck(R) to the limitands, inducing a unique (dotted) continuous linear map to the limit, as in * * k−1 k−1 k−1 C (R) ::: / C (Kn+1) / C (Kn) / ::: O 5 O f f2 O j j f f f d j j f f dx j j f f f j j f f f k fj f f k k−1 C (R) ::: / C (Kn+1) / C (Kn) / ::: 5 4 d k k−1 That is, there is a unique continuous linear map dx : C (R) ! C (R) compatible with the differentiations on finite intervals. === Similarly, 1 k 1 1 d 1 1 [1.3] Theorem: C (R) = limk C (R), also C (R) = limn C (Kn), and dx : C (R) ! C (R) is continuous. 1 k Proof: From C (R) = limk C (R) we can obtain the induced map d=dx, as follows. Starting with the commutative diagram ) * C1( ) ::: Ck−1( ) Ck−1( ) ::: R / R / R / ; d r9 d ww dx rr dx ww rr ww rr ww rrr ww C1(R) ::: / Ck(R) / Ck(R) / ::: 5 5 2 Paul Garrett: Examples of function spaces (February 11, 2017) Composing the projections with d=dx gives (dashed) induced maps from Ck(R) to the limitands, inducing a unique (dotted) continuous linear map to the limit, as in ) * C1(R) ::: / Ck−1(R) / Ck−1(R) / ::: O 5 g3 9 w; k k g g g rr ww d k g g rr w dx k g r ww k k g g rr ww k g g g rrr ww 1 gk g k k C (R) ::: / C (R) / C (R) / ::: 5 5 k A novelty is the assertion that (projective) limits commute with each other, so that the limits of C (Kn) in k and in n can be taken in either order. Generally, in a situation a _ ] [ h e c Y V j _ \ T R l f b X + ( limj(limi Vij) ::: / limi Vi2 / limi Vi1 limi(limj Vij) £¤ ¨ . . ( 1 : ( limj V2j ::: / V22 / V21 1 2 4 6 ; ? Ö Ö limj V1j ::: /2 V12 /4 V11 the maps limj(limi Vij) ! Vk` induce a map limj(limi Vij) ! lim` Vk`, which induce a unique limj(limi Vij) ! limk(lim` Vk`). Similarly, a unique map is induced in the opposite direction, and, for the usual reason, these are mutual inverses. === [1.4] Claim: For fixed x 2 R and fixed non-negative integer k, the evaluation map f ! f (k)(x) is continuous. Proof: Take n large enough so that x 2 [−n; n]. Evaluation f ! f (k)(x) was shown to be continuous on Ck[−n; n]. Composing with the continuous C1(R) ! Ck(R) ! Ck[−n; n] gives the continuity. === k k 2. Banach completion Co (R) of Cc (R) o It is reasonable to ask about the completion of the space Cc (R) of compactly-supported continuous functions k in the metric given by the sup-norm, and, more generally, about the completion of the space Cc (R) of compactly-supported k-times continuously differentiable functions in the metric given by the sum of the sups of the k derivatives. k The spaces Cc (R) are not complete with those norms, because supports can leak out to infinity: for example, in fix any u such that u(x) = 1 for jxj ≤ 1, 0 ≤ u(x) ≤ 1 for 1 ≤ jxj ≤ 2, and u(x) = 0 for jxj ≥ 2. Then 1 X u(x − n) f(x) = n2 n=0 3 Paul Garrett: Examples of function spaces (February 11, 2017) converges in sup-norm, the partial sums have compact support, but the whole does not have compact support. o [2.1] Claim: The completion of the space Cc (R) of compactly-supported continuous functions in the metric o given by the sup-norm jfj o = sup jf(x)j is the space C ( ) of continuous functions f vanishing at C x2R o R infinity, in the sense that, given " > 0, there is a compact interval K = [−N; N] ⊂ X such that jf(x)j < " for x 62 K. o o [2.2] Remark: Since we need to distinguish compactly-supported functions Cc (R) from functions Co (R) going to 0 at infinity, we cannot use the latter notation for the former, unlike some sources. o Proof: This is almost a tautology. Given f 2 Co (R), given " > 0, let K = [−N; N] ⊂ X be compact such that jf(x)j < " for x 62 K. It is easy to make an auxiliary function ' that is continuous, compactly-supported, real-valued function such that ' = 1 on K and 0 ≤ ' ≤ 1 on X.
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