arXiv:1808.05182v3 [math.FA] 4 Oct 2019 hni hr nextension an there is When irtd n[3 .0Lma .10.Wa-togprincipl Weak-strong 140]. subsets p. open Lemma, on 2.10 functions space [13, Schwartz in or Bierstedt H semi-Montel regular a completely in a values on functions continuous weighted For on oal ovxHudr pc vrtefield the over space F( Hausdorff convex locally a 5]frBnc spaces Banach for 354] salna oooia smrhs no ..t t range, its to i.e. operators into, linear isomorphism tinuous topological linear a is quasi-complete n htteei oal ovxHudr space Hausdorff convex locally a is there that and space u oGohnic,mil,i h aethat case the in mainly, Grothendieck, to due 1.1. Ω esc htfrevery for that such be .1]t ecietesaeo ekyhlmrhcrs.harm resp. holomorphic weakly Tyll of and space Laitila disc resp. the unit 243] describe p. b to 10, 14] used Lemma p. were [18, theorem in Dixmier-Ng Lindström and the on basing Linearisations eketnin.Tepeiedsrpino hspolmre problem this of description precise The extensions. weak eiMne space. semi-Montel ups httepitevaluations point the that Suppose . naraiease for answer affirmative An esuytepolmo xedn etrvle ucin v functions vector-valued extending of problem the study We 2010 Date e od n phrases. and words Key Ω Ω Question. =F( ∶= ) uhta h map the that such F XESO FVCO-AUDFNTOSAND FUNCTIONS VECTOR-VALUED OF EXTENSION coe ,2019. 7, October : ahmtc ujc Classification. Subject Mathematics ( set a hc aewa xesosi space a in extensions weak have which npriua,w pl hmt banasqec pc repre space sequence a obtain to them apply F( we Frerick, particular, Bonet, of In results extend vector-valu and of operators representation continuous a upon base obtained results ucin ihvle nalclycne asoffspace Hausdorff convex locally a in values with functions Abstract. Ω Ω Ω ) = Ω εE E , Ω , D K ofntosi etrvle counterpart vector-valued a in functions to , ) ⊂ E a ecniee salnaiaino asbpc of) subspace (a of linearisation a as considered be can rmakonrpeetto of representation known a from ) EUNESAEREPRESENTATION SPACE SEQUENCE Let C o ie ls ffnto pcswa-togprinciple weak-strong spaces function of class wider a For . oal ovxHudr pc of space Hausdorff convex locally a egv nfidapoc ohnl h rbe fextending of problem the handle to approach unified a give We ihvle na(ope)Bnc space Banach (complex) a in values with Λ S F( ∶ easbe of subset a be e ′ E xeso,vector-valued, extension, ∈ Ω F G n yGohnic n[7 hoèe1 .3-8 for 37-38] p. 1, Théorème [37, in Grothendieck by and Ω ) ( F εE Ω h function the , Λ ⊂ ) F ∈ εE → C = 1. ASE KRUSE KARSTEN ( Ω F( eesonb ufr n[5 hoe 6 p. 76, Theorem [25, in Dunford by shown were Introduction Ω ∶= and E , Ω Ω L E , rmr 64,Scnay4A3 46E10. 46A03, Secondary 46E40, Primary and δ e ) x ( 1 G of F ) eogt h dual the to belong F( u , e ( G F( = f ′ E Ω Ω ○ i.e. , Ω iersbpc of subspace linear a E F ) , z→ ǫ f K pout egt rce-cwrzspace, Fréchet-Schwartz weight, -product, eksrn rnil sgvnby given is principle weak-strong a κ ′ , ′ ( K K ∶ E , ) Ω Λ scle eksrn principle. weak-strong a called is ) fsaa-audfntoson functions scalar-valued of [ F . fra rcmlxnmesand numbers complex or real of ) x ) → F( S sncerand nuclear is Λ F( ↦ sSchwartz’ is K = Ω K Ω u f a netninin extension an has E , vle ucin naset a on functions -valued E , dfntosa linear as functions ed ( rmc n Jordá. and Gramsch δ ? hr h pc fcon- of space the where x d sflos Let follows. as ads ) ) E )] of udr space ausdorff of F( , E ncfntoso the on functions onic vrafield a over F( n[8 em 5.2, Lemma [58, in i oe,Domański Bonet, y sfrholomorphic for es . E Ω etto of sentation ateeitneof existence the ia Ω vle functions -valued ) E ε , ′ E K pout The -product. ′ o every for Let . ) opee(see complete The . K F , f ∶ ( Ω Λ F Ω x E are s with E , ( → Ω ∈ (1) be E ) ) Ω . . 2 K. KRUSE

[38, Chap. II, §3, n○3, Théorème 13, p. 80]) which covers the case that F Ω is the space C∞ Ω of smooth functions on an open set Ω ⊂ Rd (with its usual topology).( ) Gramsch( ) [33] analized the weak-strong principles of Grothendieck and realized that they can be used to extend functions if Λ is a set of uniqueness, i.e. from f ∈F Ω and f x = 0 for all x ∈ Λ follows that f = 0, and FV Ω a semi-Montel space,( E) complete( ) and G = E′ (see [33, 0.1, p. 217]). An extension( ) result for holomorphic functions where G = E′ and E is sequentially complete was shown by Bogdanowicz in [17, Corollary 3, p. 665]. Grosse-Erdmann proved in [34, 5.2 Theorem, p. 35] for holomorphic functions on Λ = Ω that it is sufficient to test locally bounded functions f with values in a locally complete space E with functionals from a weak⋆-dense subspace G of E′ Arendt and Nikolski [6], [7] shortened his proof in the case that E is a Fréchet space (see [6, Theorem 3.1, p. 787] and [6, Remark 3.3, p. 787]). Arendt gave an affirmative answer in [5, Theorem 5.4, p. 74] for harmonic functions on an open subset Λ = Ω ⊂ Rd where the range space E is a Banach space and G a weak⋆-dense subspace of E′. In [33] Gramsch also derived extension results for a large class of Fréchet-Montel spaces F Ω in the case that Λ is a special set of uniqueness, E sequentially complete and G strongly( ) dense in E′ (see [33, 3.3 Satz, p. 228-229]). He applied it to the space of holomorphic functions and Grosse-Erdmann [36] expanded this result by ⋆ ′ the case of E being Br-complete and G only a weak -dense subspace of E (see [36, Theorem 2, p. 401] and [36, Remark 2 (a), p. 406]). In a series of papers [20], [29], [30], [43], [44] these results were generalised and improved by Bonet, Frerick, Jordá and Wengenroth who used (1) to obtain extensions for vector-valued functions via extensions of linear operators. In [43], [44] by Jordá for holomorphic functions on a domain (i.e. open and connected) Ω ⊂ C and weighted holomorphic functions on a domain Ω in a Banach space. In [20] by Bonet, Frerick and Jordá for closed subsheaves F Ω of the sheaf of smooth functions C∞ Ω on a domain Ω ⊂ Rd. Their results implied( ) some consequences on the work( of) Bierstedt and Holtmanns [14] as well. Further, in [29] by Frerick and Jordá for closed subsheaves F Ω of smooth functions on a domain Ω ⊂ Rd which are closed in the sheaf C Ω of( continuous) functions and in [30] by the first two authors and Wengenroth in( the) case that F Ω is the space of bounded functions in the kernel of a hypoelliptic linear partial( differential) operator, in particular, the spaces of bounded holomorphic or harmonic functions. The results of [30] are not used in present paper but will be treated separately and extended in [56]. In this paper we present a unified approach to the extension problem for a large class of function spaces. The spaces we treat are usally of the kind that F Ω belongs to the class of semi-Montel or Fréchet-Schwartz spaces. Even quite general( ) weighted spaces F Ω are treated, at least, if E is a semi-Montel space. The case of Banach spaces( ) is handled in [30] and [56]. Our approach is based on the representation of (a subspace) of F Ω, E as a space of continuous linear operators via the map S from (1). All our examples( ) of such spaces are actually of the form of a general weighted space FV Ω, E introduced in [52] which is generated by linear operators T E on a domain in(EΩ and) equipped with a kind of graph topology (see Definition 2.4). Spaces of this form cover many examples of function spaces like the ones we already mentioned and standard examples of such spaces are weighted spaces of continuously partially differentiable functions which are generated by the partial derivative operators. The key to generalise Question 1.1 and to obtain that S is a topological isomorphism (into) lies in a condition on the interplay of S and the pair of operators T K,T E which we call consistency (see Definition 2.2 and Theorem 2.8). This condition( is) used to extend the mentioned results and we always EXTENSION 3 have to balance the sets Λ from which we extend our functions and the subspaces G ⊂ E′ with which we test. The case of ‚thin‘ sets Λ and ‚thick‘ subspaces G is handled in Section 3 and 5, the converse case of ‚thick‘ sets Λ and ‚thin‘ subspaces G in Section 4. In our last section an application of our results is given to represent the E-valued space of 2π-periodic smooth functions and the multiplier space of the Schwartz space by sequence spaces with explicit isomorphisms describing this representation (see Corollary 6.2, Corollary 6.3).

2. Notation and Preliminaries The notation and preliminaries are essentially the same as in [52, 55, Section 2, 3]. We equip the spaces Rd, d ∈ N, and C with the usual Euclidean norm ⋅ . Furthermore, for a subset M of a topological space X we denote by M the closureS S of M in X. For a subset M of a topological vector space X, we write acx M for the closure of the absolutely convex hull acx M of M in X. ( ) By E we always denote a non-trivial locally( convex) Hausdorff space (lcHs) over the field K = R or C equipped with a directed fundamental system of seminorms pα α∈A. If E = K, then we set pα α∈A ∶= ⋅ . For more details on the theory of (locally) convex spaces see [28], [42]( or) [61]. {S S} By XΩ we denote the set of maps from a non-empty set Ω to a non-empty set X and by L F, E the space of continuous linear operators from F to E where F and E are locally( convex) Hausdorff spaces. If E = K, we just write F ′ ∶= L F, K for the dual space and G○ for the polar set of G ⊂ F . If F and E are (linearly topologically)( ) isomorphic, we write F ≅ E. We denote by Lt F, E the space L F, E equipped with the locally convex topology t of uniform convergence( ) on the finite( subsets) of F if t = σ, on the absolutely convex, compact subsets of F if t = κ, on the absolutely convex, σ F, F ′ -compact subsets of F if t = µ, on the precompact (totally bounded) subsets of(F if t)= τpc and on the bounded subsets of F if t = b. We use the symbol t F ′, F for the corresponding topology on F ′. A linear subspace G of F ′ is called separating( ) if f ′ x = 0 for every f ′ ∈ G implies x = 0. This is equivalent to G being σ F ′, F -dense( (and) κ F ′, F -dense) in F ′ by the bipolar theorem. The so-called ( ) ( ) ∶ ′ ′ ε-product of Schwartz is defined by F εE = Le Fκ, E where L Fκ, E is equipped with the topology of uniform convergence on the( equicontinu) ous( subsets) of F ′. This definition of the ε-product coincides with the original one by Schwartz [72, Chap. I, §1, Définition, p. 18]. It is symmetric which means that F εE ≅ EεF . Besides the ε-product of spaces there is an ε-product of continuous linear operators as well. For locally convex Hausdorff spaces Fi, Ei and Ti ∈ L Fi, Ei , i = 1, 2, we define the ε-product T1εT2 ∈ L F1εF2, E1εE2 of the operators (T1 and)T2 by ( ) t T1εT2 u ∶= T2 ○ u ○ T1, u ∈ F1εF2, ( )( ) t ′ ′ ′ ′ where T1∶ E1 → F1, e ↦ e ○ T1, is the dual map of T1. If T1 is an isomorphism and −1 F2 = E2, then T1ε idE2 is also an isomorphism with inverse T1 ε idE2 by [72, Chap. I, §1, Proposition 1, p. 20]. For more information on the theory of ε-products see [42], [45] and [72]. Further, for a disk D ⊂ F , i.e. a bounded, absolutely convex set, the vector space FD ∶= n∈N nD becomes a normed space if it is equipped with gauge functional of D as a⋃ norm (see [42, p. 151]). The space F is called locally complete if FD is a Banach space for every closed disk D ⊂ F (see [42, 10.2.1 Proposition, p. 197]). In the introduction we already mentioned that linearisations of spaces of vector- valued functions by means of ε-products are essential for our approach. Here, one of the important questions is which spaces of vector-valued fucntions can be represented by ε-products. Let us recall some basic definitions and results from [52, 55, Section 3]. Let Ω be a non-empty set and E an lcHs. If F Ω ⊂ KΩ is an ( ) 4 K. KRUSE

′ lcHs such that δx ∈F Ω for all x ∈ Ω, then the map ( ) ∶ Ω S F Ω εE → E , u z→ x ↦ u δx , ( ) [ ( )] is well-defined and linear. 2.1. Definition (ε-into-compatible). Let Ω be a non-empty set and E an lcHs. Let Ω Ω ′ F Ω ⊂ K and F Ω, E ⊂ E be lcHs such that δx ∈F Ω for all x ∈ Ω. We call the( spaces) F Ω and( F )Ω, E ε-into-compatible if the map( ) ( ) ( ) ∶ S F Ω εE → F Ω, E , u z→ x ↦ u δx , ( ) ( ) [ ( )] is a well-defined isomorphism into. We call F Ω and F Ω, E ε-compatible if S ( ) ( ) is an isomorphism. We write SF(Ω) if we want to emphasise the dependency on F Ω . ( ) The notion of ε-compatibility was introduced in [55, 3.4 Definition, p. 7]. Next, we introduce a concept of pairs of operators T K and T E acting on F Ω and F Ω, E , respectively, whose interplay with the map S is the key to answer( ) the question( ) of linearisation of F Ω, E via ε-products and to generalise Question 1.1. ( ) 2.2. Definition (strong, consistent). Let Ω be a non-empty set and E an lcHs. Let Ω Ω ′ F Ω ⊂ K and F Ω, E ⊂ E be lcHs such that δx ∈ F Ω for all x ∈ Ω. Let K K ( ) ( ) ∶ (ωm) E∶ E ωm m∈M be a family of non-empty sets, Tm dom Tm → K and Tm dom Tm → K Ω Ω ( ωm) E E be linear with F Ω ⊂ dom Tm ⊂ K and F Ω, E ⊂ dom Tm ⊂ E for all m ∈ M. ( ) ( ) E K a) We call Tm ,Tm m∈M a consistent family for F Ω , E , in short F, E , if for every( u ∈F) Ω εE, m ∈ M and x ∈ ωm holds( ( ) ) ( ) ( ) K ∶ ○ K ′ (i) S u ∈F Ω, E and Tm,x = δx Tm ∈F Ω , (E ) ( ) K ( ) (ii) Tm S u x = u Tm,x . (E)( K) ( ) b) We call Tm ,Tm m∈M a strong family for F Ω , E , in short F, E , if for ′ ′ every e (∈ E , f ∈F) Ω, E , m ∈ M and x ∈(ωm( holds) ) ( ) (i) e′ ○ f ∈F Ω , ( ) K ′ ○ ( ) ′ ○ E (ii) Tm e f x = e Tm f x . ( )( ) ( )( ) E K As a convention we omit the index m of the set ωm, the operators Tm and Tm E K if M is a singleton. If the family Tm ,Tm m∈M is incorporated in the topology of and F Ω and F Ω, E in the sense( of a weighted) graph topology, then consistency implies( ε-into-compatibility) ( ) which we are about to explain. In this case the spaces F Ω and F Ω, E are weighted spaces whose topology is induced by a family of ( ) ( ) K E weights V and operators Tm m∈M and Tm m∈M , respectively. ( ) ( ) 2.3. Definition (weight function, [52, 3.2 Definition, p. 5]). Let J be a non-empty set and ωm m∈M a family of non-empty sets. We call V ∶= νj,m j∈J,m∈M a family ( ) ∶ ∞ ( ) of weight functions on ωm m∈M if νj,m ωm → 0, for all j ∈ J, m ∈ M and ( ) [ ) ∀ m ∈ M, x ∈ ωm ∃ j ∈ J ∶ 0 < νj,m x . (2) ( ) 2.4. Definition ([52, 3.3 Definition, p. 5]). Let Ω be a non-empty set, a family of ∶ E∶ Ω E weight functions V = νj,m j∈J,m∈M given on ωm m∈M and Tm E ⊃ dom Tm → Ω Eωm a linear map for every( )m ∈ M. Let AP Ω(, E )be a linear subspace of E and define the space of intersections ( ) ∶ ∩ E F Ω, E = AP Ω, E dom Tm ( ) ( ) ‰m∈M Ž as well as

FV Ω, E ∶= f ∈ F Ω, E ∀ j ∈ J, m ∈ M, α ∈ A ∶ f j,m,α < ∞ ( ) ™ ( ) S S S ž EXTENSION 5 where ∶ E f j,m,α = sup pα Tm f x νj,m x . S S x∈ωm ‰ ( )( )Ž ( ) Further, we write FV Ω ∶=FV Ω, K . If we want to emphasise dependencies, we write M FV or M E( instead) of( M.) We omit the index α if E is a normed space. ( ) ( ) In AP Ω, E additional properties of the functions are gathered which are not incorporated( into) the topology. It is easy to check that FV Ω, E is locally convex but need not be Hausdorff. Furthermore, we need the point( ) evaluations to be elements of the dual FV Ω ′ for the map S to be defined. ( ) 2.5. Definition (dom-space, [52, 3.4 Definition, p. 6]). We call FV Ω, E a dom- space if it is an lcHs, the system of seminorms ⋅ j,m,α j∈J,m∈M,α∈A is( directed) and, ′ in addition, δx ∈FV Ω for every x ∈ Ω if E =(SK.S ) ( ) 2.6. Definition (generator). Consider the dom-spaces FV Ω and FV Ω, E with M ∶= M K = M E . ( ) ( ) ( ) ( E) K a) We call Tm ,Tm m∈M from Definition 2.4 a generator for FV Ω , E , in short FV( , E . ) ( ( ) ) ( ) E K b) We call a generator Tm ,Tm m∈M consistent if it is consistent in the sense of Definition 2.2 a). ( ) E K c) We call a generator Tm ,Tm m∈M strong if it is strong in the sense of Definition 2.2 b). ( ) The following remark shows that the preceding definition of a consistent resp. strong generator coincides with the one given in [52, 3.8, 3.11 Definition, p. 6].

K ′ 2.7. Remark. We note that the condition Tm,x ∈FV Ω for all m ∈ M and x ∈ ωm in a)(i) of Definition 2.2 is always satisfied for generators( ) by [52, 3.7 Lemma a), ∩ E p. 7] and (2). Moreover, if S u ∈ AP Ω, E dom Tm for u ∈ FV Ω εE and all m ∈ M and a)(ii) of Definition( ) 2.2 is fulfilled,( ) then S u ∈ FV Ω, E( )by [52, 3.7 Lemma b), p. 7] implying that a)(i) is satisfied. Further,( ) if f( ∈ FV) Ω, E and ′ ○ ∩ K ′ ′ ( ) e f ∈ AP Ω dom Tm for all e ∈ E and m ∈ M and b)(ii) of Definition 2.2 is fulfilled, then( ) e′ ○ f ∈ FV Ω by [52, 3.12 Lemma, p. 10] implying that b)(i) is satisfied. ( )

E K 2.8. Theorem ([52, 3.9 Theorem, p. 9]). Let Tm ,Tm m∈M be a consistent generator for FV, E . Then S∶ FV Ω εE → FV Ω, E( is an isomorphism) into, i.e. FV Ω and(FV Ω,) E are ε-into-compatible.( ) ( ) ( ) ( ) Sufficient conditions for ε-compatibility involving the strength of the generator as well can be found in [52, 3.17 Theorem, p. 12]. Let us a give a standard example, namely, weighted spaces of continuously partially differentiable functions. More examples can be found in [52, 55]. We recall the definition of continuous partial differentiability of a vector-valued function. A function f∶ Ω → E on an open set Ω ⊂ Rd to an lcHs E is called continuously partially differentiable (f is C1) if for d the n-th unit vector en ∈ R the limit + − en E f x hen f x ∂ f x ∶= lim ( ) ( ) h→0 h ( ) ( ) h∈R,h≠0 0 exists in E for every x ∈ Ω and ∂en Ef is continuous on Ω ( ∂en Ef is C ) for every 1 ≤ n ≤ d. For k ∈ N a function( f) is said to be k-times continuously( ) partially differentiable (f is Ck) if f is C1 and all its first partial derivatives are Ck−1. A function f is called infinitely continuously partially differentiable (f is C∞) if f k is C for every k ∈ N. For k ∈ N∞ ∶= N ∪ ∞ the linear space of all functions { } 6 K. KRUSE

k k k d f∶ Ω → E which are C is denoted by C Ω, E . Let f ∈C Ω, E . For β ∈ N0 with ∶ d βn E ∶ ( ) ( ) β = n=1 βn ≤ k we set ∂ f = f if βn = 0, and S S ∑ ( ) ∂βn Ef ∶= ∂en E⋯ ∂en E f ( ) ( ) ( ) ´¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¸¹¹¹¹¹βn-times¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¶ if βn ≠ 0 as well as ∂β Ef ∶= ∂β1 E⋯ ∂βd Ef. ( ) (K ) ( ) If E = K, we usually write ∂βf ∶= ∂β f. ( ) d 2.9. Example ([52, 3.6, 3.15 Example, p. 6, 11, 28]). Let k ∈ N∞ and Ω ⊂ R be open. We consider the cases d (i) ωm ∶= Mm × Ω with Mm ∶= β ∈ N0 β ≤ min m, k for all m ∈ N0, or d (ii) ωm ∶= N0 × Ω for all m ∈ N0{and k =S∞ S S, ( )} k and let V ∶= νj,m j∈J,m∈N0 be a directed family of weights on ωm m∈N0 where directed means( that) for every j1, j2 ∈ J and m1,m2 ∈ N0 there are(j3 ∈)J, m3 ∈ N0, m3 m1,m2, and C 0 such that νj1,m1 ,νj2,m2 Cνj3 ,m3 . We define the weighted space≥ of k-times continuously> partially differentiable≤ functions with values in an lcHs E as k k CV Ω, E ∶= f ∈C Ω, E ∀ j ∈ J, m ∈ N0, α ∈ A ∶ f j,m,α < ∞ ( ) { ( ) S S S } where β E f j,m,α ∶= sup pα ∂ f x νj,m β, x . S S (β,x)∈ωm ‰( ) ( )Ž ( ) E ∶ k Setting dom Tm =C Ω, E and ( ) E∶ k ωm β E Tm C Ω, E → E , f z→ β, x ↦ ∂ f x , (3) ( ) [( ) ( ) ( )] as well as AP Ω, E ∶= EΩ, we observe that CVk Ω, E is a dom-space and ( ) E ( ) f j,m,α = sup pα Tm f x νj,m x . ∈ S S x ωm ‰ ( )Ž ( ) b) The space Ck Ω, E with its usual topology of uniform convergence of all partial derivatives up( to order) k on compact subsets of Ω is a special case of a)(i) with J ∶= K ⊂ Ω K compact , νK,m β, x ∶= χK x , β, x ∈ ωm, for all m ∈ N0 and K ∈ J{ where SχK is the characteristic} ( ) function( ) of(K.) In this case we write Wk ∶=Vk for the family of weight functions. c) The Schwartz space is defined by d ∞ d S R , E ∶= f ∈C R , E ∀ m ∈ N0, α ∈ A ∶ f m,α < ∞ ( ) { ( ) S S S } where β E 2 m~2 f m,α ∶= sup pα ∂ f x 1 + x . S S x∈Rd ‰( ) ( )Ž( S S ) d β∈N0 ,SβS≤m d This is a special case of a)(i) with k = ∞, Ω = R , J = 1 and ν1,m β, x ∶= 2 m~2 { } ( ) 1 + x , β, x ∈ ωm, for all m ∈ N0. ( S S ) ( ) d) Let K ∶= K ⊂ Ω K compact and Mp p∈N0 be a sequence of positive real { (SMp) } ( ) numbers. The space E Ω, E of ultradifferentiable functions of class Mp of Beurling-type is defined as ( ) ( )

(Mp) ∞ E Ω, E ∶= f ∈C Ω, E ∀ K ∈ K,h > 0, α ∈ A ∶ f (K,h),α < ∞ ( ) { ( ) S S S } where 1 f ∶ sup p ∂β Ef x . (K,h),α = α SβS x∈K h MSβS S S d ‰( ) ( )Ž β∈N0 EXTENSION 7

1 ∶ K × R>0 ∶ This is a special case of a)(ii) with J = and ν(K,h),m β, x = χK x hSβSM , ( ) ( ) SβS β, x ∈ ωm, for all K,h ∈ J and m ∈ N0. ( ) ( ) {Mp} e) Let K and Mp p∈N0 be as in d). The space E Ω, E of ultradifferentiable functions of class( M)p of Roumieu-type is defined as ( ) { } {Mp} ∞ E Ω, E ∶= f ∈C Ω, E ∀ K,H ∈ J, α ∈ A ∶ f (K,H),α < ∞ ( ) { ( ) S ( ) S S } where ∶ × ∃ ∞ ∀ ∶ ⋅ ⋅ J = K H = Hn n∈N hk k∈N,hk > 0,hk ↗ n ∈ N Hn = h1 ... hn { ( ) S ( ) } and β E 1 f (K,H),α ∶= sup pα ∂ f x x∈K HSβSMSβS S S d ‰( ) ( )Ž β∈N0 (see [48, Proposition 3.5, p. 675]). Again, this is a special case of a)(ii) with 1 ν(K,H),m β, x ∶= χK x , β, x ∈ ωm, for all K,H ∈ J and m ∈ N0. HSβSMSβS ( ) ( d) ( ) ∶ ( ) f) Let n ∈ N, βi ∈ N0 with βi ≤ k and ai Ω → K for 1 ≤ i ≤ n. We set S S n Ω E∶ k E ∶ βi E P ∂ C Ω, E → E , P ∂ f x = ai x ∂ f x . ( ) ( ) ( ) ( )( ) iQ=1 ( )( ) ( )( ) and obtain the (topological) subspace of CVk Ω, E given by ( ) k ∶ k E CVP (∂) Ω, E = f ∈CV Ω, E f ∈ ker P ∂ . ( ) { ( ) S ( ) } d g) In the case (i), i.e. ωm = Mm × Ω with Mm = β ∈ N0 β ≤ min m, k for all k m ∈ N0, we define the topological subspace of CV {Ω, E fromS S S a) consisting( )} of the functions that vanish with all their derivatives when( weigh) ted at infinity by k k CV0 Ω, E ∶= f ∈CV Ω, E ∀ j ∈ J, m ∈ N0, α ∈ A,ε > 0 ( ) { ( ) S ∃ K ⊂ Ω compact ∶ f Ω∖K,j,m,α < ε S S } where β E f Ω∖K,j,m,α ∶= sup pα ∂ f x νj,m β, x . S S x∈Ω∖K ‰( ) ( )Ž ( ) β∈Mm k ∶ k ∩ k Further, we define its subspace CVP (∂),0 Ω, E = CV0 Ω, E CVP (∂) Ω, E with the linear partial differential operator P (∂ E from) f). ( ) ( ) ( ) k d If V , k ∈ N∞, is locally bounded away from zero on an open set Ω ⊂ R , i.e. for every compact set K ⊂ Ω and m ∈ N0 there is j ∈ J such that

inf νj,m β, x > 0, ∈ ∈Nd x K,β 0 ( ) SβS≤min(m,k) k k then the inclusion CV Ω → CW Ω , f ↦ f, is continuous and we have the following result concerning( ) consistency,( ) strength and ε-into-compatibility by virtue of the Banach-Steinhaus theorem. k 2.10. Proposition. Let E be an lcHs, k ∈ N∞, V a directed family of weights which is locally bounded away from zero on an open set Ω ⊂ Rd and F Ω barrelled k k k k ( ) where F stands for CV , CV0, CVP (∂) or CVP (∂),0. Then the following holds. a) If u ∈F Ω εE, then S u ∈Ck Ω, E and ( β) E ( ) ( β ) d ∂ S u x = u δx ○ ∂ , β ∈ N0, β ≤ k, x ∈ Ω. ( ) ( )( ) ( ) S S b) If e′ ∈ E′ and f ∈F Ω, E , then e′ ○ f ∈Ck Ω, E and β ′ ( ′ ) β E ( d ) ∂ e ○ f x = e ∂ f x , β ∈ N0, β ≤ k, x ∈ Ω. β E (β )( ) ‰( ) ( )Ž S S c) ∂ , ∂ β∈Nd,SβS≤m with m ≤ k is a strong, consistent family for F, E . (( ) ) 0 ( ) 8 K. KRUSE

E K d) Tm ,Tm m∈N0 from (3) is a strong, consistent generator for F, E . e) F( Ω and) F Ω, E are ε-into-compatible. ( ) ( ) ( ) Proof. Part a) is shown in [52, 4.12 Proposition, p. 22] and b) in [52, 3.21 Example, p. 14]. Part c) is included in part d) by the definition of the generator. The consistency and strength for F, E in part d) is a direct consequence of a) and b) if F =CV∞. The additional properties( ) of vanishing at infinity or being in the kernel of P ∂ needed for S u in (i) of Definition 2.2 a) for u ∈F Ω εE and for e′ ○ f in ( ) ( ) ′ ′ (k ) k k (i) of Definition 2.2 b) for e ∈ E and f ∈F Ω, E if F =CV0 , CVP (∂) or CVP (∂),0 ∞ are proved in [53, 3.15 Proposition b), p. 9-10]( for)F =CV0 and in the proof of [52, ∞ ∞ 3.27 Example, p. 17] for F =CVP (∂), CVP (∂),0. Part e) follows from d) by Theorem 2.8. 

3. Extension of vector-valued functions K Using the functionals Tm,x, we extend the definition of a set of uniqueness and a space of restrictions given in [20, Definition 4, 5, p. 230]. This prepares the ground E K for a generalisation of Question 1.1 using a strong, consistent family Tm ,Tm m∈M . ( ) 3.1. Definition (set of uniqueness). Let Ω be a non-empty set, F Ω ⊂ KΩ an K ∶ ωm( ) lcHs, ωm m∈M be a family of non-empty sets and Tm F Ω → K be linear for ( ) × ( ) K all m ∈ M. U ⊂ m∈M m ωm is called a set of uniqeness for Tm, F m∈M if ∀ ⋃ ∶ { K} ′ ( ) (i) m, x ∈ U Tm,x ∈F Ω , ∀ ( ) ∶ K ( ) ∀ (ii) f ∈F Ω Tm f x = 0 m, x ∈ U ⇒ f = 0. ( ) ( )( ) K ( ) We omit the index m in ωm and Tm if M is a singleton and consider U as a subset of Ω. K K If U is a set of uniqueness for Tm, F m∈M , then span Tm,x m, x ∈ U is dense ′ ′ ( ) { S ( ) } in F Ω σ (and F Ω κ) by the bipolar theorem. ( ) ( ) 3.2. Remark. Let Ω be a non-empty set and F Ω ⊂ KΩ an lcHs. ( ) a) A simple set of uniqueness for idKΩ , F is given by U ∶= Ω. b) If F Ω has a Schauder basis f(n n∈N with) associated sequence of coefficient ( ) K ∶ K ( ) ∶ K functionals T = Tn n∈N. Then U = N is a set of uniqueness for T , F . ( ) ( ) An example for b) is the space of holomorphic functions on an open disc Dr z0 ⊂ C with radius 0 < r ≤ ∞ and center z0 ∈ C. If we equip this space with compact-( ) n open topology, then it has the shifted monomials ⋅ − z0 n∈N0 as a Schauder n (( ) ) n (n) basis with the point evaluations δz0 ○ ∂C n∈N0 given by δz0 ○ ∂C f ∶= f z0 (n) as associated sequence of coefficient( functionals) where f ( z0 denotes)( ) the n(-th) complex derivative at z0 of a holomorphic function f on Dr(z0). We will explore further sets of uniqueness for concrete function spaces in the( upcoming) examples and come back to b) in our last section. 3.3. Definition (restriction space). Let G ⊂ E′ be a separating subspace and U a K ∶ set of uniqueness for Tm, F m∈M . Let FG U, E be the space of functions f U → E (′ ) ( ) K ′ ○ such that for every e ∈ G there is fe′ ∈F Ω with Tm fe′ x = e f m, x for all m, x ∈ U. ( ) ( )( ) ( ) ( ) 3.4. Remark. Since U is a set of uniqueness, the functions fe′ are unique and the R ∶ ′ R ′ ∶ map f E → F Ω , f e = fe′ , is well-defined and linear. ( ) ( ) 3.5. Remark. Let F Ω and F Ω, E be ε-into-compatible. Consider a set of K( ) ( ) ′ uniqueness U for Tm, F m∈M , a separating subspace G ⊂ E and a strong, con- E( K ) ∶ sistent family Tm ,Tm m∈M for F, E . For u ∈ F Ω εE set f = S u . Then ( ) ( ) ( ) ( ) EXTENSION 9 f ∈F Ω, E by the ε-into-compatibility and we set f∶ U E, f m, x ∶= T E f x . ̃ → ̃ m It follows( that) ( ) ( )( ) K e′ ○ f m, x = e′ ○ T E f x = T e′ ○ f x ̃ m m ( ′) ( ( ))( )′ ′ ( )( ) for all m, x ∈ U and fe′ ∶= e ○f ∈F Ω for all e ∈ E by the strength of the family. ( ) ( ) We conclude that f ∈FG U, E . ̃ ( ) Under the assumptions of the preceding remark the map ∶ E RU,G S F Ω εE → FG U, E , f ↦ Tm f x (m,x)∈U , ( ( ) ) ( ) ( ( E)( )) is well-defined. The map RU,G is also linear since Tm is linear for all m ∈ M. Further, the strength of the defining family guarantees that RU,G is injective. 3.6. Proposition. Let F Ω and F Ω, E be ε-into-compatible, G ⊂ E′ a separating ( ) ( ) K E K subspace and U a set of uniqueness for Tm, F m∈M . If Tm ,Tm m∈M is a strong family for F, E , then the map ( ) ( ) ( ) E∶ U E T F Ω, E → E , f ↦ Tm f x (m,x)∈U , ( ) ( ( )( )) is injective, in particular, RU,G is injective. Proof. Let f ∈F Ω, E with T E f = 0. Then ( ) ( ) ′ ○ E ′ ○ E K ′ ○ 0 = e T f m, x = e Tm f x = Tm e f x , m, x ∈ U, ( )( ) ( )( ) ( )( ) ( ) and e′ ○ f ∈F Ω for all e′ ∈ E′ by the strength of the family. Since U is a set of uniqueness, we( get) that e′ ○ f = 0 for all e′ ∈ E′ which implies f = 0.  3.7. Question. Let F Ω and F Ω, E be ε-into-compatible, G ⊂ E′ a separating E K ( ) ( ) subspace, Tm ,Tm m∈M a strong family for F, E and U a set of uniqueness for K ( ) ( ) Tm, F m∈M . When is the injective restriction map ( ) ∶ E RU,G S F Ω εE → FG U, E , f ↦ Tm f x (m,x)∈U , ( ( ) ) ( ) ( ( )( )) surjective? The Question 1.1 is a special case of this question if there is a set of uniqueness K K U for Tm, F m∈M with Tm,x m, x ∈ U = δx x ∈ Λ , Λ ⊂ Ω. We observe that a positive( answer) to the{ surjectivityS ( ) of RΩ},G {resultsS in} the following weak-strong principle. 3.8. Proposition. Let F Ω and F Ω, E be ε-into-compatible, G ⊂ E′ a separating subspace such that e′ ○ f ∈F( ) Ω for( all e′)∈ G and f ∈F Ω, E . If ∶ ( ) ( ) RΩ,G S F Ω εE → FG Ω, E , f ↦ f, ( ( ) ) ( ) with the set of uniqueness Ω for idKΩ , F is surjective, then ( ) ′ ′ F Ω εE ≅F Ω, E via S and F Ω, E = f∶ Ω → E ∀ e ∈ G ∶ e ○ f ∈F Ω . ( ) ( ) ( ) { S ( )} Proof. From the ε-into-compatibility and the surjectivity of RΩ,G we obtain ∶ ∀ ′ ∶ ′ ○ f Ω → E e ∈ G e f ∈F Ω =FG Ω, E = S F Ω εE ⊂F Ω, E . { S ( )} ( ) ( ( ) ) ( ) Further, the assumption that e′ ○ f ∈F Ω for all e′ ∈ G and f ∈F Ω, E , implies that F Ω, E is a subspace of the space( ) on the left-hand side which( proves) our statement,( in) particular, the surjectivity of S.  To answer Question 3.7 for general sets of uniqueness we have to restrict to a certain class of separating subspaces of E′. 3.9. Definition (determine boundedness [20, p. 230]). A linear subspace G ⊂ E′ determines boundedness if every σ E, G -bounded set B ⊂ E is already bounded in E. ( ) 10 K. KRUSE

In [27, p. 139] such a space G is called uniform boundedness deciding by Fernán- dez et al. and in [63, p. 63] w∗-thick by Nygaard if E is a Banach space. 3.10. Remark. a) Let E be an lcHs. Then G ∶= E′ determines boundedness by [61, Mackey’s theorem 23.15, p. 268]. b) Let X be a barrelled lcHs, Y an lcHs and E ∶= Lb X, Y . For x ∈ X and ′ ′ ∶ ′ ( ∶ ) ′ y ∈ Y we set δx,y′ L X, Y → K, T → y T x , and G = δx,y′ x ∈ X, y ∈ Y ′ ⊂ E′. Then G determines( ) boundedness( ( )) (in E) by Mackey’s{ S theorem and} the uniform boundedness principle. For Banach spaces X,Y this is already observed in [20, Remark 11, p. 233] and, if in addition Y = K, in [6, Remark 1.4 b), p. 781]. c) Further examples and a characterisation of subspaces G ⊂ E′ that determine boundedness can be found in [6, Remark 1.4, p. 781-782], [63, Theorem 1.5, p. 63-64] and [63, Theorem 2.3, 2.4, p. 67-68] in the case that E is a Banach space.

F Ω a semi-Montel space and E (sequentially) complete ( ) Our next results are in need of spaces F Ω such that closed graph theorems hold with Banach spaces as domain spaces and( ) F Ω as the range space. Let us formally define this class of spaces. ( ) 3.11. Definition (BC-space [65, p. 395]). We call an lcHs F a BC-space if for every Banach space X and every linear map f∶ X → F with closed graph in X × F , one has that f is continuous. A characterisation of BC-spaces is given by Powell in [65, 6.1 Corollary, p. 400- 401]. Since every Banach space is ultrabornological and barrelled, the [61, Closed graph theorem 24.31, p. 289] of de Wilde and the Pták-K¯omura-Adasch-Valdivia closed graph theorem [50, §34, 9.(7), p. 46] imply that webbed spaces and infra-(s)- spaces are BC-spaces. Adasch calls an lcHs F an infra-(s)-space if H ∩ F ′ = F ′ for ̂ every σ F ′, F -dense subspace H of F ′ where H is the σ F ⋆, F -quasi-closure of ̂ H in the( algebraic) dual F ⋆ (see [4, Definitionen, p. 210]).( Due to) the closed graph theorems [50, §34, 9.(7), p. 46] and [76, Theorema 1, 2, p. 52, 53] the notion of an infra-(s)-space and of a Γr-space by Valdivia [76, Definicion 1, p. 52] coincide. In particular, every Br-complete space is an infra-(s)-space by [50, §34, 9.(7), p. 46]. ′ f ′ An lcHs F is called a Br-complete space if every σ F , F -dense σ F , F -closed linear subspace of F ′ equals F ′ where σf F ′, F is( the finest) topology( coinciding) with σ F ′, F on all equicontinuous sets in( F ′ (see) [50, §34, p. 26]). An lcHs F is called B( -complete) if every σf F ′, F -closed linear subspace of F ′ is weakly closed. In particular, B-complete spaces( are) Br-complete and every Br-complete space is complete by [50, §34, 2.(1), p. 26]. These definitions are equivalent to the original definitions of Br- and B-completeness by Pták [66, Definition 2, 5, p. 50, 55] due to [50, §34, 2.(2), p. 26-27] and we note that they are also called infra-Pták spaces and Pták spaces, respectively. 3.12. Proposition. The following spaces are BC-spaces: a) webbed spaces, e.g. Fréchet spaces, LF-spaces and strong duals of LF-spaces. b) infra-(s)-spaces (Γr-spaces), e.g. (i) Br-complete spaces (infra-Pták spaces), especially, B-complete spaces ′ (Pták spaces), Fréchet spaces and the dual Ft of a Fréchet space F with ′ ′ a locally convex topology τpc F , F ≤ t ≤ µ F , F , ′ ( ) ( ) ′ (ii) Ft where F is a Fréchet space with a locally convex topology σ F , F ≤ ′ t ≤ τpc F , F . ( ) ( ) EXTENSION 11

Proof. Let us start with part a). Fréchet spaces and LF-spaces are webbed spaces by [42, 5.2.2 Proposition, p. 90] and [42, 5.3.3 Corollary (b), p. 92] and strong duals of LF-spaces are webbed by [45, Satz 7.25, p. 165]. Turning to part b), we observe that Fréchet spaces are B-complete by [42, 9.5.2 ˘ ′ Krein-Smulian Theorem, p. 184] and the dual Ft of a Fréchet space F with a ′ ′ locally convex topology τpc F , F ≤ t ≤ µ F , F by [50, §34, 3.(5), p. 30] as well. ( ′ ) ( ) This implies that the dual Ft of a Fréchet space F with a locally convex topology ′ ′ σ F , F ≤ t ≤ τpc F , F is an infra-(s)-space by [50, §34, 9.(5), p. 46].  ( ) ( ) The following proposition is a modification of [45, Satz 10.6, p. 237] and uses the map Rf from Remark 3.4. K 3.13. Proposition. Let U be a set of uniqueness for Tm, F m∈M and F Ω a BC- R ○ ( ) ( ) space. Then f Bα is bounded in F Ω for every f ∈FE′ U, E and α ∈ A where ∶ ( ) ( ) ( ) R ○ Bα = x ∈ E pα x < 1 . In addition, if F Ω is semi-Montel, then f Bα is relatively{ compactS ( in) F Ω} . ( ) ( ) ( ) ○ ′ ′ Proof. Let f ∈FE′ U, E and α ∈ A. The polar B is compact in E and thus E ○ α σ Bα is a Banach space by( [61,) Corollary 23.14, p. 268]. We claim that the restriction of ′ ′ ′ ′ Rf to E ○ has closed graph. Indeed, let e be a net in E ○ converging to e in Bα τ Bα ′ R ′ ( ) EB○ and f eτ converging to g in F Ω . For m, x ∈ U we note that α ( ) ( ) ( ) K ′ K ′ ′ K R ′ ○ ○ ′ Tm,x f eτ = Tm feτ x = eτ f m, x → e f m, x = Tm fe x ‰ ( )Ž ( )( ) ( ) ( ) ( )( ) K R ′ = Tm f e x . ‰ ( )Ž( ) K K ′ The left-hand side converges to Tm,x g since Tm,x ∈ F Ω for all m, x ∈ U. K K R (′ ) ( ) ( ) Hence we have Tm g x = Tm f e x for all m, x ∈ U. From U being a ′ set of uniqueness follows( )( ) that g‰= R(f )eŽ(.) Since F (Ω is) a BC-space, we obtain ′ that the restriction of Rf to the Banach( space) E ○ is( continuous.) This yields that Bα ○ ○ ′ Rf B is bounded as B is bounded in E ○ . If F Ω is also a semi-Montel space, α α Bα ( R) ○ ( )  then f Bα is even relatively compact. ( ) Now, we are ready to prove our first extension theorem. Its proof of surjectivity of RU,E′ is just an adaption of the proof of surjectivity of S given in [52, 3.14 K Theorem, p. 9]. Let U be a set of uniqueness for Tm, F m∈M . For f ∈FE′ U, E we consider the dual map ( ) ( ) Rt ∶ ′ ′⋆ Rt ′ ∶ f F Ω → E , f y e = y fe′ , ( ) ( )( ) ( ) ′⋆ ′ ′⋆ where E is the algebraic dual of E . Further, we denote by J ∶ E → E the canonical injection.

E K 3.14. Theorem. Let F Ω and F Ω, E be ε-into-compatible, Tm ,Tm m∈M a strong, consistent family( for) F, E ,(F Ω ) a semi-Montel BC-space( and U )a set of K ( ) ( ) uniqueness for Tm, F m∈M . If ( ) (i) E is complete, or if ′ ′ (ii) E is sequentially complete and for every f ∈ FE′ U, E and f ∈ F Ω ′ ′ ( ′ ) ′ ( ) there is a sequence fn n∈N in F Ω converging to f in F Ω κ such that Rt ′ ( ) ( ) ( ) f fn ∈ J E for every n ∈ N, ( ) ( ) ∶ then the restriction map RU,E′ S F Ω εE → FE′ U, E is surjective. ( ( ) ) ( ) Proof. Let f ∈FE′ U, E . We equip J E with the system of seminorms given by ( ) ( ) ′ ○ ∶ sup pBα J x = J x e = pα x , x ∈ E, (4) ′∈ ○ ( ( )) e Bα S ( )( )S ( ) 12 K. KRUSE

∶ Rt ′ for all α ∈ A where Bα = x ∈ E pα x < 1 . We claim f ∈ L F Ω κ, J E . Indeed, we have for y ∈F Ω{ ′ S ( ) } ( ( ) ( )) t ( ) ○ R sup ′ sup sup pBα f y = y fe = y x ≤ y x (5) ′∈ ○ ∈R ○ ∈ ‰ ( )Ž e Bα S ( )S x f (Bα) S ( )S x Kα S ( )S ∶ R ○ R ○ where Kα = f Bα . Due to Proposition 3.13 the set f Bα is absolutely convex and relatively compact( ) implying that Kα is absolutely convex( and) compact in F Ω by [42, 6.2.1 Proposition, p. 103]. Further, we have for all e′ ∈ E′ and m, x ∈ U( ) ( ) Rt K ′ K ′ ○ ′ f Tm,x e = Tm,x fe′ = e f m, x = J f m, x e (6) ( )( ) ( ) ( ) ‰ ( )Ž( ) Rt K and thus f Tm,x ∈ J E . First, let condition( ) (i)( be) satisfied, i.e. E be complete, and f ′ ∈F Ω ′. The span K ′ ( ) of Tm,x m, x ∈ U is dense in F Ω κ since U is a set of uniqueness for F Ω . Thus{ thereS ( is a) net }f ′ of the form( )f ′ = nτ a T K converging to f(′ in) τ τ τ k=1 k,τ (m,x)k,τ ′ ( ) ∑ F Ω κ. As ( ) nτ Rt ′ f fτ = J ak,τ f m, x k,τ ∈ J E ( ) ‰kQ=1 (( ) )Ž ( ) and t ′ t ′ ′ ′ ○ R − R sup − 0 pBα f fτ f f ≤ fτ f x → , (7) ‰ ( ) ( )Ž (5) x∈Kα S( )( )S Rt ′ for all α ∈ A, we gain that f fτ τ is a Cauchy net in the complete space J E . ( ( )) Rt ′ ( ) Hence it has a limit g ∈ J E which coincides with f f since ( ) ( ) t ′ t ′ t ′ t ′ ○ − R ○ − R + ○ R − R pBα g f f ≤ pBα g f fτ pBα f fτ f f ‰ ( )Ž ‰ t( ′)Ž ‰ ′( )′ ( )Ž p ○ g − R f + sup f − f x 0. ≤ Bα f τ τ → (7) ‰ ( )Ž x∈KαT( )( )T Rt ′ ′ ′ We conclude that f f ∈ J E for every f ∈F Ω . Second, let condition( ) (ii) be( satisfied) and f ′ ∈F( Ω) ′. Then there is a sequence ′ ′ ′ ′ ( ) Rt ′ fn n∈N in F Ω converging to f in F Ω κ such that f fn ∈ J E for every ( ) ( ) Rt ′ ( ) ( ) ( ) n ∈ N. From (5) we derive that f fn n is a Cauchy sequence in the sequentially ( (Rt)) ′ complete space J E converging to f f ∈ J E . ( ) ( R) t ( ) ′ Therefore we obtain in both cases that f ∈ L F Ω κ, J E . So we get for all α ∈ A and y ∈F Ω ′ ( ( ) ( )) ( ) −1 ○ Rt −1 ○ R′ Rt pα J f y = pB○ J J f y = pB○ f y ≤ sup y x . (4) α α ‰( )( )Ž ‰ (( )( ))Ž ‰ ( )Ž (5) x∈Kα S ( )S −1 ○ Rt ′ ∶ This implies J f ∈ L F Ω κ, E = F Ω εE (as linear spaces). We set F = −1 ○ Rt ( ( ) ) ( ) S J f and obtain from consistency that ( ) E E −1 ○ Rt −1 Rt K −1 Tm F x = Tm S J f x = J f Tm,x = J J f m, x = f m, x ( )( ) ( )( ) ‰ ( )Ž (6) ‰ ( ( ))Ž ( ) for every m, x ∈ U which means RU,E′ F = f.  ( ) ( ) K K If E is complete and U a set of uniqueness for Tm, F m∈M with Tm,x m, x ∈ U = δx x ∈ Λ , Λ ⊂ Ω, then we get [33, 0.1, p. 217]( as) a special case.{ S ( ) }Let{ usS prepare} an application of this extension theorem to a non-Fréchet B- complete space F Ω . Let F, ⋅ be a Banach space and τ a locally convex Hausdorff topology( on) F which( isY coarserY) than the ⋅ -topology. Using the mixed topology γ τ, ⋅ introduced by Wiweger [79, p. 49],Y Y we get a large class of semi- Montel B-complete( Y Y) spaces. For the definition of transseparability, hypercomplete- ness and an S-space in the sense of Husain we use in the next proposition see [64, Definition 2.5.1, p. 53], [64, Definition 3.2.15, p. 86] and [41, Ch. 6, §1, Definition 1, p. 71], respectively. EXTENSION 13

3.15. Proposition. Let F, ⋅ be a Banach space and τ a locally convex Hausdorff topology on F which is coarser( Y Y) than the ⋅ -topology such that the closed ⋅ -unit Y Y Y Y ball BY⋅Y is τ-compact. Then F,γ is a transseparable, hypercomplete semi-Montel S-space in the sense of Husain( where) γ ∶= γ τ, ⋅ is the mixed topology. In particular, F,γ is B-complete. ( Y Y) ( ) Proof. The τ-compactness of BY⋅Y implies that F,γ is semi-Montel by [23, Section I.1, 1.13 Proposition, p. 11]. Further, [23, Section( I.1,) 1.12 Proposition, p. 10] says that a subset of F is γ-precompact if and only if it is ⋅ -bounded and τ-precompact. Y Y Hence the singleton BY⋅Y is a fundamental system of γ-precompact sets and we deduce from [41, Ch.{ 6, §6,} Proposition 5, p. 80] that F,γ is an S-space. In addition, the property that every ⋅ -bounded set is γ-precompact( ) yields to the Y Y transseparability of F,γ by [64, Proposition 2.5.3, p. 53] with t ∶= τY⋅Y and s ∶= γ. ( ) Let us turn to hypercompleteness. BY⋅Y is τ-complete by [42, 3.2.1 Proposition, p. 58] because it is τ-compact. Thus we obtain the completeness of F,γ from the ( ) τ-completeness of BY⋅Y and [23, Section I.1, 1.14 Proposition, p. 11]. Due to F,γ being a complete S-space we have that F,γ is also hypercomplete by [41, Ch.( 6,) §3, Corollary 6, p. 76]. The observation( that) every hypercomplete locally convex space is B-complete by [41, Ch. 5, §6, Corollary 5, p. 69] completes the proof. 

Let us give a concrete example of a space with mixed topology which fulfils the assumptions of the preceding proposition. For an open set Ω ⊂ Rd, an lcHs E and E ∞ ∞ a partial differential operator P ∂ ∶ C Ω, E → C Ω, E which is hypoelliptic if E = K we define the spaces of( zero) solutions( ) ( ) ∞ ∶ ∞ E CP (∂) Ω, E = f ∈C Ω, E f ∈ ker P ∂ ( ) { ( ) S ( ) } and of bounded zero solutions ∞ ∶ ∞ ∀ ∶ ∶ ∞ CP (∂),b Ω, E = f ∈CP (∂) Ω, E α ∈ A f ∞,α = sup pα f x < . ( ) { ( ) S Y Y x∈Ω ( ( )) }

Apart from the topology given by ⋅ ∞,α α∈A there is another weighted locally ∞ (Y Y ) convex topology on CP (∂),b Ω, E which is of interest, namely, the one induced by the seminorms ( ) ∶ ∞ f ν,α = sup pα f x ν x , f ∈CP (∂),b Ω, E , S S x∈Ω ( ( ))S ( )S ( ) for ν ∈C0 Ω and α ∈ A where C0 Ω is the space of K-valued continuous functions ( ) ( ) ∞ ∞ on Ω that vanish at infinity. We denote by CP (∂),b Ω, E ,β the space CP (∂),b Ω, E ( ( ) ) ⋅ ( ) equipped with the topology β induced by the seminorms ν,α ν∈C0(Ω),α∈A. The topology β is called the strict topology and we will see that(S itS coincide) s for E = K with the mixed topology γ τco, ⋅ ∞ where τco denotes the compact-open topology ∞ ( Y Y ) on CP (∂),b Ω , i.e. the topology of uniform convergence on compact subsets of Ω. ( ) 3.16. Proposition. Let Ω ⊂ Rd be open and P ∂ K a hypoelliptic linear partial ∞ ( ) differential operator. Then CP (∂),b Ω ,β is a transseparable, hypercomplete semi- Montel S-space in the sense( of Husain,( ) in) particular, B-complete. Proof. It follows from [56, 3.9 Proposition, p. 8] and the proof of [56, 3.10 Corollary, ∞ ⋅ ⋅ p. 8] that CP (∂),b Ω , ∞ is a Banach space and that the closed ∞-unit ball ∞ ⋅ ( ( ) Y Y ) Ω Y Y BY Y∞ is τco-compact in CP (∂),b . Due to [22, Proposition 3, p. 590], saying that ( ) the β-topology coincides with the mixed topology γ τco, ⋅ ∞ on the space Cb Ω of bounded continuous functions on Ω, and [23, Section( I.4,Y Y 4.6) Proposition, p.( 44],) saying that this is inherited by subspaces if BY⋅Y∞ is τco-compact, we obtain that ⋅ ∞  β = γ τco, ∞ on CP (∂),b Ω . Proposition 3.15 proves our statement. ( Y Y ) ( ) 14 K. KRUSE

3.17. Remark. Let Ω ⊂ Rd be open and P ∂ K a hypoelliptic linear partial differ- ∞ ( ) ential operator. The space CP (∂),b Ω ,β is neither bornological nor barrelled, in particular, not metrisable by( [23, Section( ) ) I.1, 1.15 Proposition, p. 12]. But it is a sequential space by [57, Proposition 5.6, p. 14], i.e. every sequentially open set in ∞ CP (∂),b Ω ,β is already open. ( ( ) ) ∞ Thus CP (∂),b Ω ,β is an example of a B-complete space which is not a Fréchet space but( still close( ) by) being a sequential space. In the case that Ω = D is the open unit disc in C and P ∂ = ∂ the Cauchy-Riemann operator the space C∞ D ,β ( ) ( ∂,b( ) ) is even separable since the monomials form a Cesàro basis by [23, Section V.2, 2.2 Proposition, p. 232]. ∞ ∞ The proof of the ε-compatibility of CP (∂),b Ω ,β and CP (∂),b Ω, E ,β is our ( ∞ ( ) ) ( ( ) ) next step. This is a bit tricky because CP (∂),b Ω ,β is not barrelled and hence we cannot use Proposition 2.10 a) directly.( ( ) ) 3.18. Proposition. Let Ω ⊂ Rd be open and P ∂ K a hypoelliptic linear partial ∞ ( ∞) differential operator. Then CP (∂),b Ω ,β εE ≅ CP (∂),b Ω, E ,β via S if E is a quasi-complete lcHs. ( ( ) ) ( ( ) ) ∞ ∶ Ω K Proof. We set AP Ω, E =CP (∂),b Ω, E and observe that idE , idΩ is the gen- ∞ ( ) ( ) ( ) erator of CP (∂),b Ω ,β , E . First, we prove that the generator is consistent. (( ( ) ) ) ∞ Clearly, we only need to show that S u ∈ AP Ω, E for every u ∈ CP (∂),b Ω ,β εE. ∞ ( ) ( ) ∞ ( ( )∞ ) Let u ∈ CP (∂),b Ω ,β εE. Next, we show that u ∈CWP (∂) Ω εE with CWP (∂) Ω ( ( ) ) ∞ ( ) ( ) from Example 2.9. For α ∈ A there are K ⊂ CP (∂),b Ω ,β compact and C > 0 such ′ ∞ ′ ( ( ) ) that for all f ∈ CP (∂),b Ω ,β holds ( ( ) ) ′ ′ pα u f ≤ C sup f f . (8) ( ( )) f∈K S ( )S ∞ ⋅ From the compactness of K in CP (∂),b Ω ,β follows that K is ∞-bounded ( ( ) ) ∞ Y Y and τco-compact by [22, Proposition 1 (viii), p. 586] since CP (∂),b Ω ,β carries ( ( ) ) the induced topology of Cb Ω ,β and the strict topology β is the mixed topology ⋅ ( ( ) ) ′ ∞ ′ γ = γ τco, ∞ by [22, Proposition 3, p. 590]. Let f ∈ CP (∂) Ω , τco . Then there( are MY ⊂Y Ω)compact and C0 > 0 such that ( ( ) ) ′ f f ≤ C0 sup f x S ( )S x∈M S ( )S ∞ ∞ for all f ∈CP (∂) Ω . Choosing a compactly supported cut-off function ν ∈ Cc Ω with ν = 1 near M( ,) we obtain ( ) ′ f f ≤ C0 sup f x ν x = C0 f ν S ( )S x∈Ω S ( )SS ( )S S S ∞ ′ ∞ ′ for all f ∈ CP (∂) Ω . Therefore f ∈ CP (∂) Ω ,β . In combination with the ( ) ( ( ) ) ∞ τco-compactness of K it follows from (8) that u ∈ CP (∂) Ω , τco εE. Using that ∞ ∞ ( ( ) ) CP (∂) Ω , τco = CWP (∂) Ω as locally convex spaces by the hypoellipticity of ( K( ) ) ( ) ∞ P ∂ (see e.g. [30, p. 690]), we obtain that u ∈CWP (∂) Ω εE. Due to Proposition ( ) ∞ ( ) 2.10 a) this yields that S u ∈CP (∂) Ω, E . Furthermore, we note that ( ) ( ) S u ∞,α = sup pα S u x = sup pα u δx ≤ C sup sup δx f Y ( )Y x∈Ω ( ( )( )) x∈Ω ( ( )) (8) x∈Ω f∈K S ( )S = C sup f ∞ < ∞ f∈K Y Y ⋅ ∞ as K is ∞-bounded implying that S u ∈CP (∂),b Ω, E = AP Ω, E . Hence the Y Y ( ) ( ) ( ) generator idEΩ , idΩK is consistent. ( ) EXTENSION 15

′ ○ ∞ ′ ′ ∞ It is easily seen that e f ∈CP (∂),b Ω = AP Ω for all e ∈ E and f ∈CP,b Ω, E by Proposition 2.10 b) which proves( that) the( generator) is strong as well.( More-) over, we observe that the set Nν,1 f ∶= f x ν x x ∈ Ω is relatively com- ∞ ( ) { ( )S ( )S S } pact in E for every f ∈ CP (∂),b Ω, E ,β and ν ∈ C0 Ω by [12, §1, 16. Lemma, p. 15] and the quasi-completeness( ( of )E.) Therefore the( ) closure of the absolutely convex hull acx Nν,1 f of Nν,1 f is compact in E by [78, 9-2-10 Example, p. 134]. As a consequence( ( )) [52, 3.16( Condition) d), p. 12] is fulfilled and we get that ∞ ∞ CP (∂),b Ω ,β εE ≅ CP (∂),b Ω, E ,β via S by [52, 3.17 Theorem, p. 12] in combi- nation( with( ) [52,) 3.13( Lemma( b), p.) 10].)  If Ω ⊂ C is an open, simply connected set, P ∂ = ∂ and E is complete, then the preceding result is also a consequence of [13, 3.10( ) Satz, p. 146]. 3.19. Corollary. Let Ω ⊂ Rd be open, E a complete lcHs and P ∂ K a hypoelliptic E K ( ) linear partial differential operator. Let Tm ,Tm m∈M be a strong, consistent family ∞ ( ) K ∞ for CP (∂),b Ω ,β , E and U a set of uniqueness for Tm, CP (∂),b Ω ,β m∈M . ((∶ ( ) ) ) ∞( ( ( ) ))′ ′ If f U → E is a function such that there is fe′ ∈ CP (∂),b Ω for each e ∈ E K ′ ○ ( ) with Tm fe′ x = e f m, x for all m, x ∈ U, then there is a unique F ∈ ∞ ( )( ) (E )( ) ( ) CP (∂),b Ω, E with Tm F x = f m, x for all m, x ∈ U. ( ) ( )( ) ( ) ( ) ∞ Proof. Due to Proposition 3.16 and Proposition 3.12 b) the space CP (∂),b Ω ,β ∞ ∞ ( ( ) ) is a semi-Montel BC-space. Moreover, CP (∂),b Ω ,β and CP (∂),b Ω, E ,β are ε-compatible by Proposition 3.18 yielding( our statement( ) ) by( Theorem( 3.14) (i)) and Proposition 3.6.  β E β m N0 ∂ , ∂ d In particular, for every ∈ the family β∈N0 ,SβS≤m is strong and ∞ (( ) ) consistent for CP (∂),b Ω ,β , E by the proof of Proposition 3.18. (( ( ) ) ) Similarly, we may apply Theorem 3.14 to the space E{Mp} Ω, E of ultradiffer- {Mp} entiable functions of class Mp of Roumieu-type. E Ω is( a projective) limit of a countable sequence of DFS-spaces{ } by [47, Theorem 2.6,( p.) 44] and thus webbed because being webbed is stable under the formation of projective and inductive lim- its of countable sequences by [42, 5.3.3 Corollary, p. 92]. Further, if the sequence

Mp p∈N0 satisfies Komatsu’s conditions (M.1) and (M.3)’ (see [47, p. 26]), then (E{Mp)} Ω is a Montel space by [47, Theorem 5.12, p. 65-66]. The spaces E{Mp} Ω and E{(Mp)} Ω, E are ε-compatible if (M.1) and (M.3)’ hold and E is complete( by) [52, 3.24 Example( ) b), p. 15-16]. Hence Theorem 3.14 (i) is applicable. 3.20. Remark. We remark that Remark 3.5 and Theorem 3.14 still hold if the map S∶ F Ω εE → F Ω, E is only an algebraic isomorphism into, i.e. an isomorphism into( of) linear spaces,( ) since the topological nature of ε-into-compatibility is not used in the proof. In particular, this means that it can be applied to the space M Ω, E of meromorphic functions on an open, connected set Ω ⊂ C with values in( an lcHs) E over C (see [21, p. 356]). The space M Ω is a Montel LF-space, thus webbed, by the proof of [35, Theorem 3 (a), p. 294-295]( ) if it is equipped with the locally convex topology τML given in [35, p. 292]. By [21, Proposition 6, p. 357] the map S∶ M Ω εE → M Ω, E is an algebraic isomorphism if E is locally complete and does( not) contain( the space) CN. Therefore we can apply Theorem 3.14 if E is complete and does not contain CN. This augments [43, Theorem 12, p. 12] where E is assumed to be locally complete with suprabarrelled strong dual and E C T ,T = idEΩ , idCΩ . ( ) ( ) F Ω a Fréchet-Schwartz space and E locally complete ( ) We recall the following abstract extension result. 16 K. KRUSE

3.21. Proposition ([20, Proposition 7, p. 231]). Let E be locally complete, Y a ′ ′ A∶ Fréchet-Schwartz space, X ⊂ Yb = Yκ dense and X → E linear. Then the follow- ing assertions are equivalent: ( ) a) There is a (unique) extension A ∈ Y εE of A. −1 ′ ′ ′ ̂ b) At Y = e ∈ E e ○ A ∈ Y determines boundedness in E. ( ) ( ) ( { S }) Next, we generalise [20, Theorem 9, p. 232] using the preceding proposition. The proof of the generalisation is simply obtained by replacing the set of uniqueness in the proof of [20, Theorem 9, p. 232] by our more general set of uniqueness. 3.22. Theorem. Let E be locally complete, G ⊂ E′ determine boundedness and E K F Ω and F Ω, E be ε-into-compatible. Let Tm ,Tm m∈M be a strong, consistent family( ) for F(, E ,)F Ω a Fréchet-Schwartz( space and) U a set of uniqueness for K ( ) ( ) ∶ Tm, F m∈M . Then the restriction map RU,G S F Ω εE → FG U, E is surjective. ( ) ( ( ) ) ( ) ∶ K ∶ Proof. Let f ∈FG U, E . We choose X = span Tm,x m, x ∈ U and Y =F Ω . A∶ ( ) {A K S ( ∶ ) } (A ) Let X → E be the linear map generated by Tm,x = f m, x . The map is ′ ′ well-defined since G is σ E , E -dense. Let e ∈ (G and) fe′ be( the) unique element K ( ) ′ ○ A K in F Ω such that Tm fe′ x = e Tm,x for all m, x ∈ U. This equation ( ) ( )( ) ( ) ( ) K ∶ ′ ○A K allows us to consider fe′ as a linear form on X (by setting fe′ Tm,x = e Tm,x ) which yields e′ ○ A ∈ F Ω for all e′ ∈ G. It follows that G (⊂ At )−1 Y implying( ) that At −1 Y determines( ) boundedness. Applying Proposition( 3.21,) ( there) is an extension( ) A(∈F) Ω εE of A and we set F ∶= S A . We note that ̂ ( ) (̂) E E A A K A K Tm F x = Tm S x = Tm,x = Tm,x = f m, x ( )( ) (̂)( ) ̂( ) ( ) ( ) for all m, x ∈ U by consistency connoting RU,G F = f.  ( ) ( ) Let us apply the preceding theorem to our weighted spaces of continuously par- tially differentiable functions and its subspaces from Example 2.9. 3.23. Corollary. Let E be locally complete, G ⊂ E′ determine boundedness, V∞ a directed family of weights which is locally bounded away from zero on an open set d d Ω ⊂ R , let F Ω be a Fréchet-Schwartz space and U ⊂ N0 × Ω a set of uniqueness β ∞ ∞ ∞ ∞ , ∂ (d ) for F β∈N0 where F stands for CV , CV0 , CVP (∂) or CVP (∂),0. Then the following( holds.) ∶ ′ a) If f U → E is a function such that there is fe′ ∈ F Ω for each e ∈ G β ′ with ∂ fe′ x = e ○ f β, x for all β, x ∈ U, then( ) there is a unique F ∈F Ω, E( )with( ∂β E)(F x )= f β, x (for all) β, x ∈ U. b) If U ⊂(Ω and) f∶ U( E) is( a) function( ) such that( e′ ○)f admits an extension → ′ fe′ ∈F Ω for every e ∈ G, then there is a unique extension F ∈F Ω, E of f. ( ) ( ) ′ ′ c) F Ω εE ≅ F Ω, E via S and F Ω, E = f∶ Ω → E ∀ e ∈ G ∶ e ○ f ∈ F(Ω) . ( ) ( ) { S ( )} Proof. In all cases V∞ is locally bounded away from zero and the Fréchet space β E β Ω ∂ , ∂ d , E F is barrelled. This implies the consistency of β∈N0 for F and the( ε)-into-compatibility of F Ω and F Ω, E by Proposition(( ) ) 2.10 c) and( e).) β E β Ω ( ) ( ∂) , ∂ d F is a Fréchet-Schwartz space and β∈N0 obviously strong as well which( implies) that part a) and its special(( case) part) b) hold by Theorem 3.22 and Proposition 3.6. Part c) follows from part b) and Proposition 3.8 since U ∶= Ω is a set of uniqueness for idKΩ , F .  ( ) Closed subspaces of Fréchet-Schwartz spaces are also Fréchet-Schwartz spaces ∞ ∞ by [61, Proposition 24.18, p. 284]. The spaces CV0 Ω and CVP (∂),0 Ω are closed ∞ ∞ ( ) ∞ ( ) subspaces of CV Ω and CVP (∂) Ω , respectively. The space CVP (∂) Ω is closed ( ) ( ) ( ) EXTENSION 17

∞ ∶ ∞ in CV Ω if there is an lcHs Y such that P ∂ SCV∞(Ω) CV Ω → Y is continuous. For example,( ) this is fulfilled if the coefficients( of) P ∂ belong( to) C Ω , in particular ( ∞) ( ) if P ∂ ∶= ∆ or ∂, with Y ∶= C Ω , τco due to V being locally bounded away d from( zero.) If ωm = Mm × Ω with( ( M) m =) β ∈ N0 β ≤ m and J is countable, ∞ then CV Ω is a Fréchet space by [53, 3.7{ Proposition,S S S p.} 7]. Conditions on the ∞ weights V∞( which) make CV Ω and its closed subspaces nuclear Fréchet spaces, in particular, Fréchet-Schwartz( spaces) can be found in [54, 3.1 Theorem, p. 12]. For d the case ωm = N0 × Ω see the references given in [54, p. 1]. ∞ The preceding corollary can be applied to the Schwartz space CV Rd ∶= S Rd and improves the ε-compatibility given in [71, Proposition 9, p. 108,( ) Théorèm( )e 1, p. 111] (E quasi-complete) and [55, 4.9 Theorem a), p. 16] (E sequentially ∞ complete). An application to the Fréchet-Schwartz space CV Ω ∶= E(Mp) Ω of ultradifferentiable functions of class Mp of Beurling-type (see( ) [47, Theorem( ) 2.6, p. 44]) also improves [48, Theorem( 3.10,) p. 678] since Komatsu’s conditions (M.0), (M.1), (M.2)’ and (M.3)’ (see [47, p. 26] and [48, p. 653]) are not needed and the condition that E is sequentially complete is weakened to local completeness. 3.24. Remark. Let V∞ be a directed family of weights which is locally bounded away from zero on an open set Ω ⊂ Rd. ∞ a) Then any dense set U ⊂ Ω is a set of uniqueness for F, idKΩ with F =CV , ∞ ∞ ∞ ( ) CV0 , CVP (∂) or CVP (∂),0 due to continuity. b) Let Ω be connected and x0 ∈ Ω. Then U ∶= en, x 1 ≤ n ≤ d, x ∈ β{( ) S Ω ∪ 0, x0 is a set of uniqueness for F, ∂ β∈N0 by the mean value theorem} {( with)}F from a). ( ) c) Let K ∶= R, d ∶= 1, Ω ∶= a,b ⊂ R, g∶ a,b → N and x0 ∈ a,b . Then U ∶= g x , x x ∈ a,b ( ∪ ) n, x0 (n ∈ N) 0 is a set of uniqueness( ) for {(β ( ) ) S ( )} {( ) S } g(x) F, ∂ β∈N0 with F from a). Indeed, if f ∈F Ω and 0 = ∂ f x for all x( ∈ a,b) , then f is a polynomial by [24, Chap.( ) 11, Theorem, p.( 53].) If, in n addition,( ) 0 = ∂ f x0 for all n ∈ N0, then the polynomial f must vanish on the whole interval( Ω.) d) Let Ω ⊂ C be connected. Then any set U ⊂ Ω with an accumulation point ∞ Ω id Ω in is a set of uniqueness for CV∂ , C by the identity theorem for holomorphic functions. ( ) e) Let Ω ⊂ C be connected and z0 ∈ Ω. Then U ∶= n,z0 n ∈ N0 is a set ∞ n ∈N {( ) S } of uniqueness for CV∂ , ∂C n 0 by local powers series expansion and the n identity theorem where( ∂C )denotes the n-th complex differential operator which is related to the real partial differential operators by

β β2 SβS 2 ∂ f z = i ∂C f z , β ∶= β1,β2 ∈ N0, z ∈ Ω (9) ( ) ( ) ( ) for all f ∈C∞ Ω (see e.g. [51, 3.4 Lemma, p. 17]). ∂ f) Let Ω ⊂ Rd be( connected.) Then any non-empty open set U ⊂ Ω is a set of ∞ uniqueness for CV∆ , idKΩ by the identity theorem for harmonic functions (see e.g. [39, Theorem( 5, p.) 218]). ∞ g) Further examples of sets of uniqueness for CV∆ , idKΩ are given in [46]. ( ) ∞ D 0 In part e) a special case of Remark 3.2 b) is used, namely, that CW∂ r z has n ( ( )) a Schauder basis with associated coefficient functionals ∂C n∈N0 where 0 < r ≤ ∞ is such that Dr z0 ⊂ Ω. In order to obtain some sets of uniqueness( ) which are more sensible w.r.t. the( ) family of weights V∞, we turn to entire and harmonic functions ∞ fulfilling some growth conditions. For a family V ∶= νj j∈N of continuous weights on Rd, a hypoelliptic linear partial differential operator( )P ∂ and an lcHs E we define the weighted space of zero solutions ( ) ∞ d ∶ ∞ d ∀ ∶ ∼ ∞ AVP (∂) R , E = f ∈ CP (∂) R , E j ∈ N, α ∈ A f j,α < ( ) { ( ) S S S } 18 K. KRUSE where ∼ ∶ f j,α = sup pα f x νj x . S S x∈Rd ( ( )) ( ) If P ∂ = ∂, d = 2 and K = C, or P ∂ = ∆ and there is 0 ≤ τ < ∞ such that ∞ ν x( =) exp − τ + 1 x , x ∈ Rd, for( all) j ∈ N, then Aτ C, E ∶= C, E is j j ∂ AV∂ ( ) ( ( )S S)τ d ∶ ∞ d ( ) ( ) the space of entire and A∆ R , E = AV∆ R , E the space of harmonic functions of exponential type τ. If τ( = 0, then) the( elements) of these spaces are also called functions of infra-exponential type. ∞ ∶ 3.25. Condition. Let V = νj j∈N be an increasing family of continuous weights d ∶ d ( ) 1 d on R . Let there be r R → 0, 1 and for any j ∈ N let there be ψj ∈ L R , ψj > 0, d and Im j j and Am j (0 such] that for any x R : ( ) ( ) ≥ ( ) > ∈ d + 1 d + α.1 supζ∈R , YζY ≤r(x) νj x ζ A j infζ∈R , YζY∞≤r(x) νI1(j) x ζ ( ) ∞ ( ) ≤ ( ) ( ) α.2 νj x A2 j ψj x νI2(j) x ( ) ( ) ≤ ( ) ( ) ( ) α.3 νj x A3 j r x νI3(j) x ( ) ( ) ≤ ( ) ( ) ( ) The preceding condition is a special case of [54, 2.1 Condition, p. 3] with Ω ∶= ∶ d ∞ ∞,∗ ∶ Ωn = R for all n ∈ N. If V fulfils Condition 3.25 and we set V = νj,m j∈N,m∈N0 ∶ d × ∶ ∞,∗ d ( ) where νj,m β ∈ N0 β ≤ m Ω, νj,m β, x = νj x , then CV R and its closed {∞,∗ RdS S S } ( ) ( ) ( ) subspace CVP (∂) for P ∂ with continuous coefficients are nuclear by [54, 3.1 Theorem, p. 12]( in combination) ( ) with [54, 2.7 Remark, p. 5] and Fréchet spaces by [53, 3.7 Proposition, p. 7]. ∞ ∞ ∶ ∶ − + 3.26. Remark. Let 0 ≤ τ < . Then V = νj j∈N given by νj x = exp τ 1 d ( ) ( ) ( ( j x , x ∈ R , fulfils Condition 3.25 by [54, 2.8 Example (iii), p. 5-6]. Further, examples)S S) of families of weights fulfilling Condition 3.25 can be found in [54, 2.8 Example, p. 5-6] and [60, 1.5 Examples, p. 205]. ∞ d Now, we can use Corollary 3.23 and these conditions to show that AVP (∂) R , E ∞ ∗ , Rd ( ) coincides as a locally convex space with CVP (∂) , E if P ∂ = ∂ or ∆ and E is locally complete which is used in the next section( as well.) ( ) ∞ 3.27. Proposition. Let E be a locally complete lcHs. If V fulfils Condition ∞ C ∞ Rd ∞ C 3.25, then AV∂ and AV∆ are nuclear Fréchet spaces and AV∂ , E = ∞,∗ ( ) ∞ d ( ) ∞,∗ d ( ) CV C, E and AV∆ R , E = CV∆ R , E as locally convex spaces. ∂ ( ) ( ) ( ) Proof. Let P ∂ ∶= ∂ (d ∶= 2 and K ∶= C) or P ∂ ∶= ∆. First, we show that ∞ ∗ ∞ Rd ( ) , Rd ( ) ∞ Rd AVP (∂) = CVP (∂) as locally convex spaces which implies that AVP (∂) ∞ ∗ ( ) ( ) , Rd ∞ C ( N) is a nuclear Fréchet space as CVP (∂) is such a space. Let f ∈ AV∂ , j ∈ , 2 ( ) ( ) m ∈ N0, z ∈ C and β ∶= β1,β2 ∈ N0. Then it follows from ⋅ ∞ ≤ ⋅ and Cauchy’s inequality that ( ) Y Y S S β ! ∂βf z ν z iβ2 ∂SβSf z ν z sup f w ν z j = C j ≤ S S SβS j S ( )S ( )(9) S ( )S ( ) r z Sw−zS=r(z) S ( )S ( ) ( ) ≤ β !C j, β sup f w νB3 (j) z (α.3)S S ( S S) Sw−zS=r(z) S ( )S ( )

≤ β !C j, β A1 B3 j sup f w νI1B3(j) w (α.1)S S ( S S) ( ( )) Sw−zS=r(z) S ( )S ( ) ∼ ≤ β !C j, β A1 B3 j f I1B3(j) S S ( S S) ( ( ))S S where C j, β ∶= A3 j A3 I3 j ⋯A3 B3 − 1 j and B3 − 1 is the β − 1 -fold ( S S) ( ) ( ( )) (( )( )) (S S ) composition of I3. Choosing k ∶= maxSβS≤m I1B3 j , it follows that ( ) ∼ ∞ f j,m ≤ sup β !C j, β A1 B3 j f k < S S SβS≤m S S ( S S) ( ( ))S S EXTENSION 19

∞ ∗ ∞ ∞ ∗ and thus f ∈ , C and C = , C as locally convex spaces. In the CV∂ AV∂ CV∂ case P ∂ = ∆ an analogous( ) proof( works) due( to) Cauchy’s inequality for harmonic ( ) ∞ d d d functions, i.e. for all f ∈ AV∆ R , j ∈ N, x ∈ R and β ∈ N0 holds ( ) SβS β d β ∂ f x νj x ≤ S S sup f w νj x S ( )S ( ) Šr x  Sw−xS

τ 3.28. Remark. The following sets U ⊂ C are sets of uniqueness for A , idCC . ( ∂ ) a) If τ < π, then U ∶= N0 is a set of uniqueness by [15, 9.2.1 Carlson’s theorem, p. 153]. b) Let δ > 0 and λn n∈N ⊂ 0, ∞ such that λn+1 − λn > δ for all n ∈ N. Then ∶ ( ) ( ) −2τ~π ∞ U = λn n∈N is a set of uniqueness if lim supr→∞ r ψ r = where ∶( ) −1 ( ) ψ r = exp λn 0, by [15, 9.5.1 Fuchs’s theorem, p. 157-158]. ( ) (∑ ) τ n The following sets U are sets of uniqueness for A , ∂C n∈N0 . ( ∂ ) c) Let λn n∈N0 ⊂ C with λn < 1 for all n ∈ N0. If τ < ln 2 , then U ∶= n,( λn ) n ∈ N0 is a setS ofS uniqueness by [15, 9.11.1 Theorem,( ) p. 172]. If {( ) S } τ < ln 2 + √3 , then U ∶= 2n + 1, 0 n ∈ N0 ∪ 2n, λn n ∈ N0 is a set of uniqueness( ) by [15, 9.11.3{( Theorem,) S p. 173].} {( ) S } −1 n −1 ∶ d) Let λn n∈N0 ⊂ C with lim supn→∞ n k=1 λk ≤ 1. If τ < e , then U = n,( λn ) n ∈ N0 is a set of uniqueness∑ by [15,S S 9.11.4 Theorem, p. 173]. {( ) S }d τ The following sets U ⊂ R are sets of uniqueness for A∆, idRRd . ( ) e) Let d ∶= 2. If there is k ∈ N with τ < π k, then U ∶= Z ∪ Z + ik is a set of uniqueness by [16, Theorem 1, p. 425].~ ( ) f) Let d ∶= 2. If τ < π and θ ∉ πQ, then U ∶= Z ∪ eiθZ is a set of uniqueness by [16, Theorem 2, p. 426]. ( ) g) If τ < π, then U ∶= 0, 1 × Zd−1 is a set of uniqueness by [67, Corollary 1.8, p. 312]. { } −1 h) If τ < π and a ∈ R with a ≤ »1 d − 1 , then U ∶= Zd × 0,a is a set of uniqueness by [80, TheoremS S A, p.~( 335].) { } i) Further examples of sets of uniqueness can be found in [8]. τ β R U A , ∂ d The following sets are sets of uniqueness for ∆ β∈N0 . ( ( d)−1) j) If τ < π, then U ∶= β, x, 0 β ∈ 0,ed , x ∈ Z is a set of uniqueness by [80, Theorem B,{( p. 335].( )) Further S { examples} can be} found in [8]. 20 K. KRUSE

We close this section by an examination of the space ∞ k E E0 E ∶= f ∈ C 0, 1 , E ∀ k ∈ N0 ∶ ∂ f cont. extendable on 0, 1 ( ) { (( ) ) S ( ) [ ] and ∂k Ef 1 = 0 ( ) ( ) } k E k E where ∂ f 1 ∶= limx↗1 ∂ f x and which we equip with the system of semi- norms( given) by( ) ( ) ( ) ∶ k E f m,α = sup pα ∂ f x , f ∈ E0 E , S S x∈(0,1) ‰( ) ( )Ž ( ) k∈N0,k≤m for m ∈ N0 and α ∈ A. We need the following weak-strong principle in our last section. 3.29. Corollary. Let E be a locally complete lcHs and G ⊂ E′ determine bounded- ′ ′ ness. Then E0εE ≅ E0 E via S and E0 E = f∶ 0, 1 → E ∀ e ∈ G ∶ e ○ f ∈ E0 . ( ) ( ) { ( ) S } Proof. Analogously to the proof of [52, 5.11 Example, p. 29] we may deduce that k E k ∂ , ∂ k∈N0 is a strong, consistent generator for E0, E since E0 is a Fréchet- ((Schwartz) space) by [61, Example 28.9 (5), p. 350], in particul( ar,) barrelled. Therefore E0 and E0 E are ε-into-compatible by Theorem 2.8 and we derive our statement from Theorem( ) 3.22 and Proposition 3.8 with U ∶= 0, 1 .  ( ) 4. Extension of locally bounded functions In order to obtain an affirmative answer to Question 3.7 for general separating subspaces of E′ we have to restrict to the spaces F V from Definition 2.4 and a certain class of sets of uniqueness. × 4.1. Definition (fix the topology). Let FV Ω be a dom-space. U ⊂ m∈M m ωm fixes the topology in FV Ω if for every j(∈ )J and m ∈ M there are⋃ i ∈ J{, k} ∈ M and C > 0 such that ( ) K f j,m ≤ C sup Tk f x νi,k x , f ∈ Ω . ∈ FV S S x ωk S ( )( )S ( ) ( ) (k,x)∈U In particular, U is a set of uniqueness if it fixes the topology. The present definition of fixing the topology is a generalisation of [20, Definition 13, p. 234]. Sets that fix the topology appear under several different notions. Rubel and Shields call them dominating in [68, 4.10 Definition, p. 254] in the context of bounded holomorphic functions. In the context of the space of holomorphic functions with the compact-open topology studied by Grosse-Erdmann [36, p. 401] they are said to determine locally uniform convergence. Ehrenpreis [26, p. 3,4,13] (cf. [70, Definition 3.2, p. 166]) refers to them as sufficient sets when he considers inductive limits of weighted spaces of entire resp. holomorphic functions, including the case of Banach spaces. In the case of Banach spaces sufficient sets coincide with weakly sufficient sets defined by Schneider [70, Definition 2.1, p. 163] (see e.g. [49, §7, 1), p. 547]) and these notions are extended beyond spaces of holomorphic functions by Korobe˘ınik [49, p. 531]. Seip [74, p. 93] uses the term sampling sets in the context of weighted Banach spaces of holomorphic functions whereas Beurling uses the term balayage in [11, p. 341] and [11, Definition, p. 343]. Leibowitz [59, Exercise 4.1.4, p. 53], Stout [75, 7.1 Definition, p. 36] and Globevnik [32, p. 291-292] call them boundaries in the context of subalgebras of the algebra C Ω, C of complex-valued continuous functions on a compact Hausdorff space Ω with( sup-norm.) Fixing the topology is also connected to the notion of frames used by Bonet et al. in [19]. Let us set ∶ ∶ ∀ ∶ ∞ ℓV U, E = f U → E j ∈ J, m ∈ M, α ∈ A f j,m,α < ( ) { S Y Y } EXTENSION 21 with

f j,m,α ∶= sup pα f m, x νj,m x ∈ Y Y x ωm ( ( )) ( ) (m,x)∈U for an lcHs E and a set U which fixes the topology in FV Ω . If U is countable, U U the inclusion ℓV U ↪ K continuous where K is equipped( ) with the topology of pointwise convergence( ) and ℓV U contains the space of sequences (on U) with ( ) K compact support as a linear subspace, then Tk,x (k,x)∈U is an ℓV U -frame in the sense of [19, Definition 2.1, p. 3]. ( ) ( ) 4.2. Definition (lb-restriction space). Let Ω be a dom-space, U fix the topol- ′ FV ogy in FV Ω and G ⊂ E a separating subspace.( ) We set ( ) NU,i,k f ∶= f k, x νi,k x x ∈ ωk, k, x ∈ U ( ) { ( ) ( ) S ( ) } for i ∈ J, k ∈ M and f ∈ FVG U, E and ( ) ∶ ∀ ∶ FVG U, E lb = f ∈ FVG U, E i ∈ J, k ∈ M NU,i,k f bounded in E ( ) { ( ∩ ) S ( ) } =FVG U, E ℓV U, E . ( ) ( ) Consider a set U which fixes the topology in FV Ω , a separating subspace ′ E K ( ) G ⊂ E and a strong, consistent family Tm ,Tm m∈M for FV, E . For u ∈ FV Ω εE ∶ ( ) ( ) ( ) set f = S u ∈ FV Ω, E by Theorem 2.8. Then we have RU,G f ∈ FVG U, E with f ∶= S( u) by Remark( ) 3.5 and for i ∈ J and k ∈ M ( ) ( ) ( ) E ∞ sup pα y = sup pα Tk f x νi,k x ≤ f i,k,α < x∈ω y∈NU,i,k(RU,G(f)) ( ) k ( ( )( )) ( ) S S (k,x)∈U for all α ∈ A implying the boundedness of NU,i,k RU,G f in E. Thus RU,G f ∈ ( ( )) ( ) FVG U, E lb and the injective linear map ( ) ∶ E RU,G S FV Ω εE →FVG U, E lb, f ↦ Tm f x (m,x)∈U , ( ( ) ) ( ) ( ( )( )) is well-defined.

′ E K 4.3. Question. Let G ⊂ E be a separating subspace, Tm ,Tm m∈M a strong, con- sistent generator for FV, E and U fix the topology( in FV) Ω . When is the injective restriction map( ) ( ) ∶ E RU,G S FV Ω εE →FVG U, E lb, f ↦ Tm f x (m,x)∈U , ( ( ) ) ( ) ( ( )( )) surjective? ′ If G ⊂ E determines boundedness and U fixes the topology in FV Ω , then the preceding question and Question 3.7 coincide. ( )

′ E K 4.4. Remark. Let G ⊂ E determine boundedness, Tm ,Tm m∈M a strong, consis- tent generator for FV, E and U fix the topology in( FV Ω). Then ( ) ( ) FVG U, E lb = FVG U, E . ( ) ( ) Proof. We only need to show that the inclusion ⊃ holds. Let f ∈ FVG U, E . Then K ′ ○ ( ) there is fe′ ∈ FV Ω with Tm fe′ x = e f m, x for all m, x ∈ U and ( ) ′ ( )( ) (′ )( ) ( ) sup e y = sup e ○ f k, x νi,k x ≤ fe′ i,k < ∞ x∈ω y∈NU,i,k(f) S ( )S k S( )( )S ( ) S S (k,x)∈U for each e′ ∈ G, i ∈ J and k ∈ M. Since G ⊂ E′ determines boundedness, this means  that NU,i,k f is bounded in E and hence f ∈ FVG U, E lb. ( ) ( ) 22 K. KRUSE

FV Ω arbitrary and E a semi-Montel space ( ) 4.5. Definition (generalised Schwartz space). We call an lcHs E a generalised Schwartz space if every bounded set in E is already precompact. In particular, semi-Montel spaces and Schwartz spaces are generalised Schwartz spaces by [42, 10.4.3 Corollary, p. 202]. Conversely, a generalised Schwartz space is a Schwartz space if it is quasi-normable by [42, 10.7.3 Corollary, p. 215].

4.6. Proposition. Let E be an lcHs, FV Ω a dom-space and U fix the topology R ′ (R ) ○ in FV Ω . Then f ∈ L Eb, FV Ω and f Bα is bounded in FV Ω for every ( ) ( ( )) ∶ ( ) ( )R f ∈ FVE′ U, E lb and α ∈ A where Bα = x ∈ E pα x < 1 and f is the map from( Remark) 3.4. In addition, if E is{ a generalisedS ( ) Schwartz} space, then R ′ Ω R ○ Ω f ∈ L Eτpc , FV and f Bα is relatively compact in FV . ( ( )) ( ) ( ) Proof. Let f ∈ FVE′ U, E lb, j ∈ J and m ∈ M. Then there are i ∈ J, k ∈ M and C > 0 such that for every( e)′ ∈ E′ R ′ K f e j,m = fe′ j,m ≤ C sup Tk fe′ x νi,k x ∈ S ( )S S S x ωk S ( )( )S ( ) (k,x)∈U ′ ′ = C sup e ○ f k, x νi,k x = C sup e y x∈ω k S( )( )S ( ) y∈NU,i,k(f) S ( )S (k,x)∈U which proves the first part because NU,i,k f is bounded in E. Let us consider the second part. The bounded set NU,i,k f is( already) precompact in E because E is ( ) R ′ Ω a generalised Schwartz space. Therefore we have f ∈ L Eτpc , FV . The polar ○ ′ A ( ( )) Bα is relatively compact in Eτpc for every α ∈ by the Alaoğlu-Bourbaki theorem R ○  and thus f Bα in FV Ω as well. ( ) ( ) E K 4.7. Theorem. Let E be a semi-Montel space, Tm ,Tm m∈M a strong, consistent generator for FV, E , and U fix the topology in (FV Ω .) Then the restriction map ∶ ( ) ( ) RU,E′ S FV Ω εE →FVE′ U, E lb is surjective. ( ( ) ) ( ) ′ ′ ′ ′ Proof. Let f ∈ FVE′ Ω, E lb and e ∈ E . For every f ∈ FV Ω there are j ∈ J, m ∈ M and C0 > 0 with( ) ( ) Rt ′ ′ ′ f f e = f fe′ ≤ C0 fe′ j,m. S ( )( )S S ( )S S S By the proof of Proposition 4.6 there are i ∈ J, k ∈ M and C > 0 such that Rt ′ ′ ′ ′ f f e ≤ C0C sup e y ≤ C0C sup e y . S ( )( )S y∈NU,i,k(f) S ( )S y∈acx(NU,i,k(f)) S ( )S

The set acx NU,i,k f is absolutely convex and compact by [42, 6.2.1 Proposition, p. 103] and( [42, 6.7.1( )) Proposition, p. 112] because E is semi-Montel. Therefore Rt ′ ′ ′ f f ∈ Eκ = J E by the Mackey-Arens theorem. Like in Theorem 3.14 ( ) ( −1) ○ Rt ( ) ∶ we obtain J f ∈ FV Ω εE by (4), (5) and Proposition 4.6. Setting F = −1 ○ Rt ( E) S J f we conclude Tm F x = f m, x for all m, x ∈ U by (6) and so ( ) ( )( ) ( ) ( ) RU,E′ F = f.  ( ) We denote by Cbu Ω, E the space of bounded uniformly continuous functions from a metric space Ω( to an) lcHs E which we endow with the system of seminorms given by ∶ f α = sup pα f x , f ∈ Cbu Ω, E , S S x∈Ω ( ( )) ( ) for α ∈ A. EXTENSION 23

4.8. Corollary. Let Ω be a metric space, U ⊂ Ω a dense subset and E a semi-Montel ∶ ′ ○ space. If f U → E is a function such that e f admits an extension fe′ ∈ Cbu Ω ′ ′ ( ) for each e ∈ E , then there is a unique extension F ∈ Cbu Ω, E of f. In particular, ( ) ∶ ∀ ′ ′ ∶ ′ ○ Cbu Ω, E = f Ω → E e ∈ E e f ∈ Cbu Ω . ( ) { S ( )} Proof. idEΩ , idKΩ is a strong, consistent generator for Cbu, E and Cbu Ω εE ≅ ( ) ( ) ( ) Cbu Ω, E via S by [52, 5.8 Example, p. 27]. Due to Theorem 4.7, Proposition 3.6 and( Remark) 4.4 with G = E′ the extension F exists and is unique because the dense  set U ⊂ Ω fixes the topology in Cbu Ω . The rest follows from Proposition 3.8. ( ) Let Ω ⊂ C be open and bounded and E an lcHs over C. We denote by A Ω, E the space of continuous functions from Ω to an lcHs E which are holomorphic( on) Ω and equip A Ω, E with the system of seminorms given by ( ) ∶ f α = sup pα f x , f ∈ A Ω, E , S S x∈Ω ( ( )) ( ) for α ∈ A.

4.9. Corollary. Let Ω ⊂ C be open and bounded, U ⊂ Ω fix the topology in A Ω and E a semi-Montel space over C. If f∶ U E is a function such that e′ ○ f admits( ) an ′ ′ → extension fe′ ∈ A Ω for each e ∈ E , then there is a unique extension F ∈ A Ω, E of f. In particular,( ) ( ) ′ ′ ′ A Ω, E = f∶ Ω → E ∀ e ∈ E ∶ e ○ f ∈ A Ω . ( ) { S ( )} Proof. idEΩ , idCΩ is a strong, consistent generator for A, E and A Ω εE ≅ ( ) ( ) ( ) A Ω, E via S by [13, 3.1 Bemerkung, p. 141]. Due to Theorem 4.7, Proposi- tion( 3.6) and Remark 4.4 with G = E′ the extension F exists and is unique. The remaining part follows from Proposition 3.8. 

If Ω ⊂ C is connected, then the boundary ∂Ω of Ω fixes the topology in A Ω by the maximum principle. If Ω = D, then ∂D is the intersection of all sets that( fix) the topology in A D by [75, 7.7 Example, p. 39]. If E is a generalised( ) Schwartz space which is not a semi-Montel space, we do not know whether the extension results in Corollary 4.8 and Corollary 4.9 hold but we still have a weak-strong principle due to the following observation. 4.10. Proposition. Let E be a generalised Schwartz space and G ⊂ E′ determine boundedness. Then the initial topology of E and the topology σ E, G coincide on the bounded sets of E. ( ) Proof. W.l.o.g. we may assume that E is a semi-Montel space. Otherwise we use the fact that the completion E of a generalised Schwartz space is a semi-Montel space ̂ by [42, 3.5.1 Theorem, p. 64] and induces the same topology as E on a bounded set of E. Now, the proof of [40, Chap. 3, §9, Proposition 2, p. 231] with σ E, E′ replaced by σ E, G implies our statement. ( ) ( ) We only apply this observation to the space Cbu Ω, E . ( ) 4.11. Remark. If Ω is a metric space, E a generalised Schwartz space and G ⊂ E′ determine boundedness, then ∶ ∀ ′ ∶ ′ ○ Cbu Ω, E = f Ω → E e ∈ G e f ∈ Cbu Ω . ( ) { S ( )} ∶ Indeed, let us denote the right-hand side by Cbu Ω, E σ and set Eσ = E, σ E, G . ( ) ( ( )) Then Cbu Ω, E σ = Cbu Ω, Eσ and f Ω is bounded for every f ∈ Cbu Ω, E σ as G determines( boundedness.) ( Since) E is a( generalised) Schwartz space and(G determine) 24 K. KRUSE boundedness, the initial topology of E and σ E, G coincide on the bounded range ( ) f Ω of f ∈ Cbu Ω, E σ by Proposition 4.10. Hence we deduce that ( ) ( ) Cbu Ω, E σ = Cbu Ω, Eσ = Cbu Ω, E . ( ) ( ) ( ) In this way Bierstedt proves his weak-strong principles for weighted continuous functions in [13, 2.10 Lemma, p. 140] with G = E′.

FV Ω a Fréchet-Schwartz space and E locally complete ( ) 4.12. Definition ([20, Definition 12, p. 8]). Let Y be a Fréchet space. An increasing ′ ○ sequence Bn n∈N of bounded subsets of Yb fixes the topology in Y if Bn n∈N is a fundamental( system) of zero neighbourhoods of Y . ( ) ′ 4.13. Remark. Let Y be a Banach space. If B ⊂ Yb is bounded, i.e. bounded w.r.t. the operator norm, such that B fixes the topology in Y , i.e. B○ is bounded in Y , then B is called an almost norming subset. Examples of almost norming subspaces are given in [6, Remark 1.2, p. 780-781]. For instance, the set of point evaluations ∶ ∶ ∞ ∶ ∞ B = δ1~n n ∈ N is almost norming for the Hardy space Y = H D = C D . { S } ( ) ∂,b( ) 4.14. Definition (chain-structured). Let FV Ω be a dom-space. We say that U ⊂ m∈N m × ωm is chain-structured if ( ) ⋃ { } (i) k, x ∈ U ⇒ ∀ m ∈ N,m k ∶ m, x U, (∀ ) ≥ ∶ ( K )∈ K (ii) k, x U, m k, f FV Ω Tk f x = Tm f x . ( )∈ ≥ ∈ ( ) ( )( ) ( )( ) d ∞ 4.15. Remark. Let Ω ⊂ R be open and V be a directed family of weights. K ∞ Concerning the operators Tm m∈N0 of CV Ω from Example 2.9 where ωm = β ∈ d × ( d)× ( ) ∞ { N0 β ≤ m Ω resp. ωm = N0 Ω, we have for all k ∈ N0 and f ∈ CV Ω that S S S }K β K d ( ) Tk f β, x = ∂ f x = Tm f β, x , β ∈ N0, β ≤ k, x ∈ Ω, ( )( ) ( ) ( )( ) S S for all m ∈ N0, m k. Hence condition (ii) of Definition 4.14 is fulfilled for any × ≥ U m∈N0 m ωm in this case. Condition (i) says that once a ‚link‘ k,β,x belongs⊂ ⋃ to U{ for} some order k, then the ‚link‘ m,β,x belongs to U for any( higher) order m as well. ( )

4.16. Definition (diagonally dominated, increasing). We say that a family V ∶= νj,m j,m∈N of weights on Ω is diagonally dominated and increasing if ωm ⊂ ωm+1 ( ) for all m ∈ N and νj,m ≤ νmax(j,m),max(j,m) on ωmin(j,m) for all j, m ∈ N as well as νj,j ≤ νj+1,j+1 on ωj for all j ∈ N.

4.17. Remark. Let Ω be a dom-space, U ⊂ m∈N m × ωm chain-structured, ′ FV G ⊂ E a separating subspace( ) and V diagonally dominated⋃ { } and increasing. a) If U fixes the topology in FV Ω , then ( ) ∀ ∶ FVG U, E lb = f ∈ FVG U, E i ∈ N NU,i f bounded in E ( ) { ( ) S ( ) } with NU,i f ∶= NU,i,i f . ( ) ( ) ∶ b) Let FV Ω be a Fréchet space. We set Um = m, x ∈ U x ∈ ωm and ∶ (j ) K ⋅ {( ′) S } Bj = m=1 Tm,x νm,m x m, x ∈ Um ⊂ FV Ω for j ∈ N. Then U { ( ) ( ) S ( ) } ( ) fixes the⋃ topology in FV Ω in the sense of Definition 4.1 if and only if the ( ) sequence Bj j∈N fixes the topology in FV Ω in the sense of Definition 4.12. ( ) ( ) Proof. Let us begin with a). We only need to show that the inclusion ’⊃‘ holds. Let f be an element of the right-hand side and i, k ∈ N. We set m ∶= max i, k and observe that for k, x ∈ U we have m, x ∈ U by (i) and ( ) ( ) ( ) ′ ○ K K ′ ○ e f k, x = Tk fe′ x = Tm fe′ x = e f m, x ( )( ) ( )( ) (ii) ( )( ) ( )( ) EXTENSION 25 for each e′ ∈ G with (i) and (ii) from the definition of U being chain-structured. Since G is separating, it follows that f k, x = f m, x . Hence we get for all α ∈ A ( ) ( ) sup pα y = sup pα f k, x νi,k x ≤ sup pα f k, x νm,m x x∈ω (i) x∈ω y∈NU,i,k(f) ( ) k ( ( )) ( ) m ( ( )) ( ) (k,x)∈U (m,x)∈U

= sup pα f m, x νm,m x < ∞ ∈ (ii) x ωm ( ( )) ( ) (m,x)∈U using that ωk ⊂ ωm and V is diagonally dominated. Let us turn to part b). ’⇒‘: Let j ∈ N and A ⊂ FV Ω be bounded. Then ( ) K ∞ sup sup y f = sup sup Tm f x νm,m x ≤ sup sup f m,m < 1≤ ≤ 1≤ ≤ y∈Bj f∈A S ( )S m j f∈A S ( )( )S ( ) f∈A m j S S (m,x)∈Um ′ since A is bounded implying that Bj is bounded in FV Ω b. Further, Bj is increasing by definition. Additionally, for all j ∈ N ( ) ( ) j ○ K Bj = f ∈ Ω sup Tm f x νm,m x ≤ 1 FV ∈ m=1{ ( ) S x ωm S ( )( )S ( ) } (m,x)∈U K = f ∈ Ω sup Tj f x νj,j x ≤ 1 FV ∈ { ( ) S x ωj S ( )( )S ( ) } (j,x)∈U ○ because U is chain-structured and V increasing. Thus Bj is a fundamental system ( ) of zero neighbourhoods of FV Ω if U fixes the topology. ’⇐‘: Let j, m ∈ N. Then there( are) i ∈ N and ε > 0 such that ○ ∶ εBi ⊂ f ∈ FV Ω f j,m ≤ 1 = Dj,m { ( ) S S S } which follows from fixing the topology in the sense of Definition 4.12. Let f ∈ Dj,m and set ∶ K f Ui = sup Ti f x νi,i x . S S (i,x)∈Ui S ( )( )S ( ) f ○ f If f ≠ 0, then ∈ Bi and thus ε ∈ Dj,m implying SfSUi SfSUi 1 f 1 f f ε f . j,m = Ui j,m ≤ Ui S S ε S S T f Ui T ε S S S S 1  The inequality f j,m ≤ ε f Ui still holds if f = 0. S S S S 4.18. Theorem ([20, Theorem 16, p. 236]). Let Y be a Fréchet-Schwartz space, A∶ ∶ Bj j∈N fix the topology in Y and X = span j∈N Bj → E be a linear map which (is bounded) on each Bj. If (⋃ ) At −1 ′ a) Y is dense in Eb and E locally complete, or if (At)−1( ) ′ b) Y is dense in Eσ and E is Br-complete, ( ) ( ) then A has a (unique) extension A ∈ Y εE. ̂ Now, we generalise [20, Theorem 17, p. 237].

′ E K 4.19. Theorem. Let E be an lcHs, G ⊂ E a separating subspace, Tm ,Tm m∈M be a strong, consistent family for FV, E , FV Ω a Fréchet-Schwartz( space,) V diagonally dominated and increasing( and U)be chain-structured( ) and fix the topology in FV Ω . If ( ) ′ a) G is dense in Eb and E locally complete, or if b) E is Br-complete, ∶ then the restriction map RU,G S FV Ω εE →FVG U, E lb is surjective. ( ( ) ) ( ) 26 K. KRUSE

∶ Proof. Let f ∈ FVG U, E lb. We set X = span j∈N Bj with Bj from Remark 4.17 b) and Y ∶= FV Ω .( Let )A∶ X → E be the linear(⋃ map determined) by ( ) A K ⋅ ∶ Tm,x νm,m x = f m, x νm,m x , ( ( ) ( )) ( ) ( ) for 1 ≤ m ≤ j and m, x ∈ Um with Um from Remark 4.17 b). The map A is ′ well-defined since G(is σ )E , E -dense, and bounded on each Bj because A Bj = j ′ ( ) ( ) m=1 NU,m f . Let e ∈ G and fe′ be the unique element in FV Ω such that ⋃K ( ) ′ ○ K ( ) Tm fe′ x = e f m, x for all m, x ∈ U which implies Tm fe′ x νm,m x = ′ ○(A )(K ) ⋅ ( ) ( ) ( )( ) ( ) e Tm,x νm,m x for all m, x ∈ Um. This equation allows us to consider fe′ ( ( ) ( )) ( K ) ⋅ ∶ ′ ○ A K ⋅ as a linear form on X (by fe′ Tm,x νm,m x = e Tm,x νm,m x ) which ′ ′( ( ) ( )) ( t −1 ( ) ( )) yields e ○ A ∈ FV Ω for all e ∈ G. It follows that G ⊂ A Y . Noting that G is σ E′, E -dense,( we) apply Theorem 4.18 and obtain an( extension) ( ) A ∈ Ω εE ̂ FV of A(. We) set F ∶= S A and observe that for all m, x ∈ U there is j ∈ N, (j )m, ̂ such that j, x Uj (and) νj,j x 0 by (2) and because( ) U is chain-structured≥ and V diagonally( dominated)∈ and( increasing.) > Due to the proof of Remark 4.17 a) we have f j, x = f m, x and thus ( ) ( ) 1 E E A A K A K ⋅ Tm F x = Tm S x = Tm,x = Tm,x νj,j x ( )( ) (̂)( ) ̂( ) νj,j x ̂( ( ) ( )) 1 ( ) A K ⋅ = Tj,x νj,j x = f j, x = f m, x νj,j x ̂( ( ) ( )) ( ) ( ) ( ) by consistency connoting RU,G F = f.  ( ) In particular, condition a) is fulfilled if E is semireflexive. Indeed, if E is semire- flexive, then E is quasi-complete by [69, Chap. IV, 5.5, Corollary 1, p. 144] and b(E′,E) µ(E′,E) G = G = E′ by [42, 11.4.1 Proposition, p. 227] and the bipolar theorem. ∞ For instance, condition b) is satisfied if E is a Fréchet space or E = CP (∂),b Ω ,β K ( ( ) ) with a hypoelliptic linear partial differential operator P ∂ and open Ω ⊂ Rd which is a Br-complete non-Fréchet space by Proposition 3.16( ) and Remark 3.17 or E is more general a space with mixed topology from Proposition 3.15. As stated, our preceding theorem generalises [20, Theorem 17, p. 237] where ∞ d FV Ω is a closed subspace of CW Ω for open, connected Ω ⊂ R . A characteri- ( ) ( ) ∞ Ω sation of sets that fix the topology in the space CW∂ of holomorphic functions on an open, connected set Ω ⊂ C is given in [20, Remark( 14,) p. 235]. The characteri- sation given in [20, Remark 14 (b), p. 235] is still valid and applied in [20, Corollary ∞ K 18, p. 238] for closed subspaces of CWP (∂) Ω where P ∂ is a hypoelliptic linear partial differential operator which satisfies( the) maximum( ) principle, namely, that U ⊂ Ω fixes the topology if and only if there is a sequence Ωn n∈N of relatively ( ) compact, open subsets of Ω with n∈N Ωn = Ω such that ∂Ωn ⊂ U ∩ Ωn+1 for all n ∈ N. Among the hypoelliptic operators⋃ P ∂ K satisfying the maximum principle are the Cauchy-Riemann operator ∂ and the( ) Laplacian ∆. Further examples can be found in [31, Corollary 3.2, p. 33]. The statement of [20, Corollary 18, p. 238] for the space of holomorphic functions is itself a generalisation of [36, Theorem 2, p. 401] with [36, Remark 2 (a), p. 406] where E is Br-complete and of [43, Theorem 6, ′ p. 10] where E is semireflexive. The case that G is dense in Eb and E is sequentially complete is covered by [33, 3.3 Satz, p. 228-229], not only for spaces of holomorphic functions, but for several classes of function spaces. Let us turn to other families of weights than ∞. Due to Proposition 3.27 we W ∞ ∗ ∞ already know that U ∶= 0 × C fixes the topology in , C = C and CV∂ AV∂ ∶ × d ∞,∗ { d} ∞ d ∞ ( ) ( ) U = 0 R in CV∆ R = AV∆ R if V fulfils Condition 3.25. Next, we concentrate{ } on the first case( ) since smaller( ) sets that fix the topology are known. EXTENSION 27

′ ∞ 4.20. Corollary. Let E be an lcHs over C, G ⊂ E a separating subspace, V fulfil C ∞ C Condition 3.25 and U ⊂ fix the topology of AV∂ . If ′ ( ) a) G is dense in Eb and E locally complete, or if b) E is Br-complete, ∶ ∞ ′ ○ and f U → E is a function in ℓV U such that e f admits an extension fe′ ∈ ∞ C ′ ( ) ∞ C AV∂ for each e ∈ G, then there is a unique extension F ∈ AV∂ , E of f. ( ) ( ) Proof. The existence of F follows from the proof of Proposition 3.27 and Theo- E C ∶ C C rem 4.19 with Tm ,Tm m∈M = idE , idC . The uniqueness of F is a result of Proposition 3.6.( ) ( )  ∞ We have the following sufficient conditions on a family of weights V which C ∞ C guarantee the existence of a countable set U ⊂ that fixes the topology of AV∂ due to Abanin and Varziev. ( ) ∞ ∶ ∶ − 4.21. Proposition. Let V = νj j∈N where νj z = exp aj µ z ϕ z , z ∈ C, with some continuous, subharmonic( ) function µ∶ C( →) 0, ∞ (, a continuous( ) ( )) function ∶ [ ) ∶ ϕ C → R and a strictly increasing, positive sequence aj j∈N with a = limj→∞ aj ∈ 0, ∞ . Let there be ( ) ( (i)] s 0 and C 0 such that ϕ z − ϕ ζ C and µ z − µ ζ C for all z,≥ ζ C with >z − ζ 1 + z S −(s,) ( )S ≤ S ( ) ( )S ≤ q (ii) max∈ ϕ z ,µ Sz S ≤z ( + CS0S)for some q, C0 0 and (iii) ln z( (= )O µ( z)) ≤as S S z → ∞ if a = ∞ or >ln z = o µ z as z → ∞ if 0 <(SaS)< ∞. ( ( )) S S (S S) ( ( )) S S ∞,1 Let λk k∈N be the sequence of simple zeros of a function L ∈ C having no AV∂ ( ) ∞,1 2 ( ) ∶ N N other zeros where V = νj νmj j∈ for some sequence mj j∈ in N. Suppose ( ~ ) ( ) ∞ that there are j0 ∈ N and a sequence of circles z ∈ C z = Rm with Rm ↗ such that { S S S }

L z νj0 z Cm, m N,z C, z = Rm, ∞ S ( )S ( ) ≥ ∈ ∈ S S for some Cm ↗ and ′ L λk νj0 λk 1 for all sufficiently large k N. S ( )S ( ) ≥ ∈ ∞ ∞ ∶ Then V fulfils Condition 3.25 for all a 0, and U = λk k∈N fixes the topology ∞ C ∞ ∈ ( ] ( ) C of AV∂ if a = . If µ is a radial function, i.e. µ z = µ z , z ∈ , with 2 ( ) ∞ (∞ )C (S S) 0 ∞ µ z µ z as z → , then U fixes the topology of AV∂ for all a , . ( ) ∼ ( ) S S ( ) ∈ ( ] Proof. First, we check that Condition 3.25 is satisfied. We set k ∶= max s, 2 and observe that (i) is also fulfilled with k instead of s. Let z ∈ C and ζ ∞( , η) ∞ ≤ −k 1 √2 1 + z =∶ r z . From ⋅ ≤ √2 ⋅ ∞ and (i) follows Y Y Y Y ( ~ )( S S) ( ) S S Y Y µ z + ζ − µ z + η ≤ µ z + ζ − µ z + µ z − µ z + η ≤ C S ( ) ( )S S ( ) ( )S S ( ) ( )S and thus µ z +ζ ≤ C +µ z +η . In the same way we obtain −ϕ z +ζ ≤ C −ϕ z +η . Hence we have( ) ( ) ( ) ( )

aj µ z + ζ − ϕ z + ζ ≤ C aj + 1 + aj µ z + η − ϕ z + η ( ) ( ) ( ) ( ) ( ) for j ∈ N implying C(aj +1) νj z + ζ ≤ e νj z + η ( ) ( ) which means that α.1 holds. By (iii) there are ε > 0 and R > 0 such that ln z ≤ εµ z for all z ∈ C (with) z R if a = ∞. This yields for all z max 2, R that(S S) ( ) S S ≥ 2 S S ≥ ( ) aj µ z + k ln 1 + z ajµ z + k ln z = aj µ z + 2k ln z ≤ ajµ z + 2kεµ z . ( ) ( S S) ≤ ( ) (S S ) ( ) (S S) ( ) ( ) Since a = ∞, there is n ∈ N such that an aj + 2kε resulting in ≥ ajµ z + k ln 1 + z anµ z ( ) ( S S) ≤ ( ) 28 K. KRUSE for all z max 2, R . Therefore we derive S S ≥ ( )

aj µ z + k ln 1 + z anµ z + k ln 1 + max 2, R (10) ( ) ( S S) ≤ ( ) ( ( )) for all z C which means that α.2 and α.3 hold with ψj z ∶= r z . If 0 < a < ∞, for every∈ ε > 0 there is R > 0 such( ) that ln( z) ≤ εµ z for( all)z ∈ C( with) z R by (iii). Thus we may choose ε 0 such that a(Sj+S)1 − aj ( 2)kε 0 because aj SisS strictly ≥ increasing. We deduce that> (10) with n ∶= j + 1 holds≥ in this> case as well( ) and α.2 and α.3 , too. ( ) Observing( ) that the condition that U = λk k∈N is the sequence of simple zeros ∞ 1 of a function L ∈ , C means that L ∈(L )Φa ; U and (i) that ϕ and µ vary AV∂ ϕ,µ slowly w.r.t. r z = 1 +(z )−s in the notation of( [3, Definition,) p. 579, 584] and [3, p. 585], respectively,( ) ( theS statementS) that U fixes the topology is a consequence of [3, Theorem 2, p. 585-586]. 

4.22. Remark. a) Let D ⊂ C be convex, bounded and open with 0 ∈ D. Let ∶ ∶ ϕ z = HD z = supζ∈D Re zζ , z ∈ C, be the supporting function of D, µ(z) ∶= ln 1(+ z) , z ∈ C, and a(j ∶=) j, j ∈ N. Then ϕ and µ fulfil the conditions of( Proposition) ( S S) 4.21 with a = ∞ by [3, p. 586] and the existence of an entire function L which fulfils the conditions of Proposition 4.21 is guaranteed by [2, Theorem 1.6, p. 1537]. Thus there is a countable set U ∶= λk k∈N ⊂ C −∞ ∶ ∞ ∞ ∶ ( −) which fixes the topology in AD = AV C with V = exp ajµ ϕ j∈N. ∂ ( ) ( ( )) b) An explicit construction of a set U ∶= λk k∈N ⊂ C which fixes the topology −∞ ( ) in AD is given in [1, Algorithm 3.2, p. 3629]. This construction does not rely on the entire function L. In particular (see [19, p. 15]), for D ∶= D 2 we have ϕ z = z , for each k ∈ N we may take lk ∈ N, lk > 2πk , and set λk,j ∶= krk,j( ,)1 ≤S jS ≤ lk, where rk,j denote the lk-roots of unity. Ordering λk,j in a sequence of one index appropriately, we obtain a sequence which −∞ fixes the topology of AD . c) Let µ∶ C → 0, ∞ be a continuous, subharmonic, radial function which increases with[ z )and satisfies S S + + + (i) supζ∈C,YζY ≤r(z) µ z ζ ≤ infζ∈C,YζY∞≤r(z) µ z ζ C for some contin- ∞ ( ) ( ) uous function r∶ C → 0, 1 and C > 0, 2 (ii) ln 1 + z = o µ z ( as ]z → ∞, (iii) µ (2 z S =S O) µ (z (S S))as z →S S∞. ( ∞S S)∶ ( (S− S)) S S Then V = exp 1 j µ j∈N fulfils Condition 3.25 where α.1 follows from (i) and ( α.2( , ( α.~ 3) ))as in the proof of Proposition 4.21.( If)µ z = 2 2 o z as z →( ∞)or( µ z) = z , z ∈ C, then U ∶= αn + iβm n,m(S ∈S)Z (S S ) S S 0(S∶ S) S∞S C 0{ S } fixes the topology in Aµ = AV∂ for any α, β > by [19, Corollary 4.6, p. 20] and [19, Proposition 4.7, p.( 20],) respectively. d) For instance, the conditions on µ in c) are fulfilled for µ z ∶= z γ , z ∈ C, with 0 < γ ≤ 2 by [60, 1.5 Examples (3), p. 205]. If γ = 1,( then) AS0 S= A0 C µ ∂ is the space of entire functions of exponential type zero (infra-exponential( ) type). e) More general characterisations of countable sets that fix the topology of ∞ C AV∂ can be found in [3, Theorem 1, p. 580] and [19, Theorem 4.5, p. 17]. ( )

0 −∞ The spaces Aµ from c) are known as Hörmander algebras and the space AD C considered in a) is isomorphic to the strong dual of the Korenblum space A−∞ (D) via Laplace transform by [62, Proposition 4, p. 580]. ( ) EXTENSION 29

5. Extension of sequentially bounded functions In this section we restrict to the case that E is a Fréchet space and G ⊂ E′ is generated by a sequence that fixes the topology in E.

5.1. Definition (sb-restriction space). Let E be a Fréchet space, Bn fix the ∶ ( ) topology in E and G = span n∈N Bn . Let FV Ω be a dom-space, U a set of uniqueness for FV Ω and set(⋃ ) ( ) ∶ ( ) ∀ ∶ ′ FVG U, E sb = f ∈ FVG U, E n ∈ N fe′ e ∈ Bn is bounded in FV Ω . ( ) { ( ) S { S } ( )} Let E be a Fréchet space, Bn fix the topology in E, G ∶= span n∈N Bn , E K ( ) (⋃ ) Tm ,Tm m∈M be a strong, consistent generator for FV, E and U a set of unique- ( ) K ( ) ness for FV Ω ,Tm m∈M . For u ∈ FV Ω εE we have RU,G f ∈ FVG U, E with f ∶= S u ( by( Remark) ) 3.5 and for j ∈ J and( ) m ∈ M ( ) ( ) ( ) ′ E ′ sup fe′ j,m = sup sup e Tm f x νj,m x = sup sup e y ′∈ ′∈ ∈ ′∈ e Bn S S e Bn x ωm S ( ( )( ) ( ))S e Bn y∈Nj,m(f) S ( )S ∶ E with Nj,m f = Tm f x νj,m x x ∈ ωm . This set is bounded in E since ( ) { ( )( ) ( ) S } sup pα f = f j,m,α < ∞ y∈Nj,m(f) ( ) S S

A sup ′ ′ ∞ for all α ∈ implying e ∈Bn fe j,m < and RU,G f ∈ FVG U, E sb. Hence the injective linear map S S ( ) ( ) ∶ E RU,G S FV Ω εE →FVG U, E sb, f ↦ Tm f x (m,x)∈U , ( ( ) ) ( ) ( ( )( )) is well-defined.

5.2. Question. Let E be a Fréchet space, Bn fix the topology in E and G ∶= E K ( ) span n∈N Bn . Let Tm ,Tm m∈M a strong, consistent generator for FV, E and (⋃ ) ( ) K ( ) U a set of uniqueness for FV Ω ,Tm m∈M . When is the injective restriction map ( ( ) ) ∶ E RU,G S FV Ω εE →FVG U, E sb, f ↦ Tm f x (m,x)∈U , ( ( ) ) ( ) ( ( )( )) surjective? 5.3. Remark. Let E be a Fréchet space with increasing system of seminorms ○ E K ∈N ∶ ∶ 1 ∈ pαn n , Bn = Bαn where Bαn = x ∈ E pαn x < , Tm ,Tm m M a strong, ( ) { S ( ) } ( ) K consistent generator for FV, E and U a set of uniqueness for FV Ω ,Tm m∈M . If ( ) ( ( ) ) (i) FV Ω is a BC-space, or if (ii) U fixes( ) the topology of FV Ω , ( ) then FVE′ U, E sb = FVE′ U, E by Proposition 3.13 in (i) resp. Remark 4.4 and Proposition( 4.6 in) (ii). Hence( Theorem) 3.14 (i) resp. Theorem 4.7 answers Question 5.2 in these cases. Let us turn to the case where G need not coincide with E′.

5.4. Proposition ([29, Lemma 9, p. 504]). Let E be a Fréchet space, Bn fix the ′ ′ ( ) topology in E, Y a Fréchet-Schwartz space and X ⊂ Yb = Yκ a dense subspace. Set ∶ A∶ ( ) G = span n∈N Bn and let X → E be a linear map which is σ X, Y -σ E, G - t continuous(⋃ and satisfies) that A Bn is bounded in Y for each n ∈(N. Then) ( A has) a (unique) extension A ∈ Y εE. ( ) ̂ Next, we improve [29, Theorem 1 ii), p. 501].

5.5. Theorem. Let E be a Fréchet space, Bn fix the topology in E and G ∶= E K ( ) span n∈N Bn , Tm ,Tm m∈M a strong, consistent generator for FV, E , FV Ω (⋃ ) ( ) ( K ) ( ) a Fréchet-Schwartz space and U a set of uniqueness for FV Ω ,Tm m∈M . Then ∶ ( ( ) ) the restriction map RU,G S FV Ω εE →FVG U, E sb is surjective. ( ( ) ) ( ) 30 K. KRUSE

∶ K ∶ Proof. Let f ∈ FVG U, E sb. We set X = span Tm,x m, x ∈ U and Y = FV Ω . A∶ ( ) { A SK ( ∶ ) } ( ) Let X → E be the linear map determined by Tm,x = f m, x which is well- defined since G is σ E′, E -dense. From ( ) ( ) ( ) ′ A K ′ ○ K e Tm,x = e f m, x = Tm,x fe′ ( ( )) ( ) ( ) for every e′ ∈ G and m, x ∈ U follows that A is σ X, Y -σ E, G -continuous and ( ) t ′ ( ) ( ) sup A e j,k = sup fe′ j,k < ∞ ′ ′ e ∈Bn S ( )S e ∈Bn S S for all j ∈ J, k ∈ M and n ∈ N. Due to Proposition 5.4 there is an extension A ∈ FV Ω εE of A. We set F ∶= S A and get for all m, x ∈ U that ̂ ( ) (̂) ( ) E E A A K Tm F x = Tm S x = Tm,x = f m, x ( )( ) (̂)( ) ̂( ) ( ) by consistency which means RU,G F = f.  ( ) 5.6. Corollary. Let E be a Fréchet space, Bn fix the topology in E and G ∶= ∞ ( ) d span n∈N Bn . Let V fulfil Condition 3.25 and U ⊂ R be a set of uniqueness for ∞(⋃ ) ∶ ′ ○ AVP (∂), idKRd where P ∂ = ∂ or ∆. If f U → E is a function such that e f ( ) ( ) ∞ d ′ ′ admits an extension fe′ ∈ AVP (∂) R for each e ∈ G and fe′ e ∈ Bn is bounded ∞ d ( ) { S }∞ d in AVP (∂) R for each n ∈ N, then there is a unique extension F ∈ AVP (∂) R , E of f. ( ) ( ) ∞ d Proof. AVP (∂) R is a Fréchet-Schwartz space and idERd , idKRd a strong, con- ( ) ∞ ( ) sistent generator for AVP (∂), E by Proposition 3.27 and the proof of Corollary 3.23. Now, Theorem 5.5( and Proposition) 3.6 prove our statement.  We already mentioned examples of families of weights ∞ that fulfil Condition ∞ V 3.25 and sets of uniqueness for AVP (∂), idKRd in Remark 3.26, Remark 3.28 and Remark 4.22. If E is a Banach space,( then an almost) norming set fixes the topology and examples can be found via Remark 4.13.

6. Representation by sequence spaces Our last section is dedicated to the representation of weighted spaces of E- valued functions by weighted spaces of E-valued sequences if there is a counterpart of this representation in the scalar-valued case involving the coefficient functionals associated to a Schauder basis (see Remark 3.2 b)). 6.1. Theorem. Let E be locally complete, G ⊂ E′ determine boundedness and Ω ′ F and F Ω, E resp. ℓ N and ℓ N, E be ε-into-compatible with e ○ g ∈ ℓ N (for) ′ ( ′ ) ( ) ( )E K ( ) all e ∈ E and g ∈ ℓ N, E . Let Tm ,Tm m∈N be a strong, consistent family for F, E and F Ω have( an equicontinuous) ( Schauder) basis with associated coefficient ( ) (K ) functionals Tm m∈N such that ( ) K∶ K ∶ K T F Ω → ℓ N ,T f = Tn f n∈N, ( ) ( ) ( ) ( ( )) is an isomorphism. If (i) Ω is a Fréchet-Schwartz space, or if F ′ (ii) E(is) sequentially complete, G = E and F Ω is a semi-Montel BC-space, then the following holds. ( ) a) FG N, E = ℓ N, E . b) ℓ N( and)ℓ N(, E are) ε-compatible, in particular, ℓ N εE ≅ ℓ N, E . c) The( ) map ( ) ( ) ( ) E∶ E ∶ E T F Ω, E → ℓ N, E ,T f = Tn f n∈N, ( ) ( ) ( ) ( ( )) is a well-defined isomorphism, F Ω and F Ω, E are ε-compatible, in par- ( E) (○ K) ○ −1 ticular, F Ω εE ≅ F Ω, E , and T = Sℓ(N) T ε idE SF(Ω). ( ) ( ) ( ) EXTENSION 31

K Proof. a) First, we remark that N is a set of uniqueness for T , F . Let u ∈ F Ω εE and n ∈ N. Then ( ) ( ) E ○ E K ○ K RN,G SF(Ω) u n = T SF(Ω) u n = Tn SF(Ω) u = u Tn = u δn T ( ( ))( ) ( K t )( )( ) K ( ( )) ( ) ( ) = u ○ T δn = T ε idE u δn ( ( ) )(K ) ( )( )( ) = Sℓ(N) ○ T ε idE u n (11) ‰ ( )Ž( )( ) by consistency and the ε-into-compatibility yielding FG N, E ⊂ ℓ N, E . Let g ∈ ′ ○ ′ ′ ∶ K −(1 ′ ○) ( ) ℓ N, E . Then e g ∈ ℓ N for all e ∈ E and ge′ = T e g ∈ F Ω . We note ( )K ( ) ( ) ( ) ( ) that Tn ge′ = ge′ n for all n ∈ N which implies ℓ N, E ⊂ FG N, E proving a). ( ) ( ) ( ) ( ) b) We only need to show that Sℓ(N) is surjective. Let g ∈ ℓ N, E which implies ( ) g ∈ FG N, E by part a). We( claim) that RN,G is surjective. In case (i) this follows directly from Theorem 3.22. Let us turn to case (ii) and denote by fn n∈N the equicontinuous Schauder K ( ) basis of Ω associated to Tn n∈N. We check that condition (ii) of Theorem 3.14 F ′ ′ is fulfilled.( Let) f ∈ F Ω and( set) ( ) k ′ ∶ ′ ∶ K ′ fk F Ω → K, fk f = Tn f f fn , ( ) ( ) nQ=1 ( ) ( ) ′ ′ ′ ′ ′ for k ∈ N. Then fk ∈ F Ω for every k ∈ N and fk converges to f in F Ω σ since k K ( ) ( ) ( ) n=1 Tn f fn converges to f in F Ω . From the equicontinuity of the Schauder (∑ ( ) ) ′ ( ) ′ ′ basis we deduce that fk converges to f in F Ω κ by [42, 8.5.1 Theorem (b), p. ( ) ′ ′ ( ) 156]. Let f ∈ FE′ N, E . For each e ∈ E and k ∈ N we have ( ) k k Rt ′ ′ ′ K ′ ′ ○ ′ f fk e = fk fe′ = Tn fe′ f fn = e f n f fn ( )( ) ( ) nQ=1 ( ) ( ) nQ=1( ( )) ( ) k ′ ′ = e f n f fn (nQ=1 ( ) ( )) Rt ′ since f ∈ FE′ N, E implying f fk ∈ J E . Hence we can apply Theorem 3.14 ( ) ( ) ( ) (ii) and obtain that RN,E′ is surjective. Thus there is u ∈ F Ω εE such that RN,E′ SF(Ω) u = g in both cases. Then K ( ) ( ( )) T ε idE u ∈ ℓ N εE and from (11) we derive ( )( ) ( ) K Sℓ(N) T ε idE u = RN,G SF(Ω) u = g (( )( )) ( ( )) connoting the surjectivity of Sℓ(N). c) First, we note that map T E is well-defined. Indeed, we have e′ ○ T E f = K ′ ′ ′ T e ○ f ∈ ℓ N for all f ∈ F Ω, E and e ∈ E by the strength of the family.( Part) ( ) ( ) E ( ) E a) implies that T f ∈ FG N, E = ℓ N, E and thus the map T is well-defined ( ) ( ) ( ) E and its linearity follows from the linearity of the Tn for n ∈ N. Next, we prove that E K T is surjective. Let g ∈ ℓ N, E . Since T ε idE is an isomorphism and Sℓ(N) by ( ) ∶ K −1 ○ −1 part b) as well, we obtain that u = T ε idE Sℓ(N) g ∈ F Ω εE. Therefore (( ) )( ) ( ) SF(Ω) u ∈ F Ω, E and from (11) we get ( ) ( ) E E K T SF(Ω) u = T ○ SF(Ω) u = Sℓ(N) ○ T ε idE u = g ( ( )) ( )( ) ‰ ( )Ž( ) which means that T E is surjective. The injectivity of T E by Proposition 3.6, implies that E −1 K SF(Ω) = T ○ Sℓ(N) ○ T ε idE ( ) ‰ ( )Ž yielding the surjectivity of SF(Ω) and thus the ε-compatibility of F Ω and F Ω, E . E ○ K ○ −1 ( ) E ( ) Furthermore, we have T = Sℓ(N) T ε idE SF(Ω) resulting in T being an isomorphism. ( )  32 K. KRUSE

Let us demonstrate an application of the preceding theorem to Fourier expan- sions of vector-valued 2π-periodic smooth functions and the multiplier space of the ∞ d Schwartz space. We equip the space C R , E for an lcHs E with the system of seminorms generated by ( ) ∶ β E ∞ d f K,m,α = sup pα ∂ f x χK x , f ∈ C R , E , x∈Ω S S d (( ) ( )) ( ) ( ) β∈N0 ,SβS≤m d ∞ d for K ⊂ R compact, m ∈ N0 and α ∈ A, i.e. we consider CW R , E . By ∞ d ( ) C2π R , E we denote its topological subspace consisting of the functions which are(2π-periodic) in each variable. If E is a locally complete lcHs over C, then −i⟨n,x⟩ d the function given by x ↦ f x e Rd is Pettis-integrable on −π, π for every ∞ d d ( ) ⋅[ ⋅ ] f ∈ C2π R , E and n ∈ Z by [55, 4.7 Lemma, p. 14] where ⟨ , ⟩Rd is the usual scalar product( ) on Rd. Hence we are able to define the n-th Fourier coefficient of ∞ d f ∈ C2π(R , E) by the Pettis-integral − − ∶ 2 d i⟨n,x⟩Rd d Zd f̂(n) = ( π) S f(x)e x, n ∈ , [−π,π]d if E is locally complete. Our aim is to prove that the map f ↦ (f̂(n))n∈Zd is an ∞ d d isomorphism from C2π(R , E) to the space s(Z , E) of rapidely decreasing E-valued sequences given by ∶ Ωd ∀ ∶ ∶ + 2 j~2 ∞ s(Ω, E) = {x = (xn) ∈ E j ∈ N, α ∈ A x j,α = sup pα xn 1 n < , S S S n∈Ω ( )( S S ) } where Ω = Zd or Nd. ∞ d ∞ d 6.2. Corollary. If E is a locally complete lcHs over C, then C2π R , E ≅ C2π R εE and s Zd, E ≅ s Zd εE and the map ( ) ( ) ( ) ( E) ∞ d d E F ∶ 2 R , E s Z , E , F f ∶= f ∈Zd , C π → ̂n n (E ) ( C ) ( −)1 ( ) F d ○ F ○ is an isomorphism with = Ss(Z ) ε idE SC∞ (Rd). ( ) 2π ∞ d ∞ d Proof. By the proof of [55, 4.10 Theorem, p. 17] the spaces C2π R and C2π R , E are ε-compatible. Moreover, the spaces s Zd and s Zd, E are( ε)-into-compatible( ) by [52, 3.9 Theorem, p. 9] and it is obvious( that) e′ ○ x( ∈ s Z)d for every e′ ∈ E′ and d ∞ d ( ) x ∈ s Z , E . The space C2π R is a Fréchet space, in particular, barrelled and − ⋅ thus its( Schauder) basis e i⟨n,(⟩Rd ) is equicontinuous. The corresponding coefficient C C functionals are given by( δn ○ F ) and the map F is an isomorphism (see e.g. [45, Satz 1.7, p. 18]). Again,( we derive) from the proof of [55, 4.10 Theorem, p. 17] that ○ E ○ C ∞ the family δn F ,δn F n∈Zd is strong and consistent for C2π, E . Now, we can apply Theorem( 6.1 yielding) our statement. ( )  The preceding corollary improves a special case of [55, 4.2 Theorem, p. 10] and [55, 4.11 Theorem, p. 19] from sequential complete E to locally complete E. In the same way we can prove the corresponding result for the Schwartz space S Rd, E d and s N0, E with sequentially complete E which is given in [55, 4.9 Theorem( a), p.) ∞ 16] by( a different) proof. For the Fréchet-Schwartz space CW DR 0 , E , 0 < R ≤ ∞ ∂ ∞, of holomorphic functions and the Köthe space λ AR, E (with( Köthe) ) matrix ∶ j ( ) AR = rk j∈N0,k∈N for some strictly increasing sequence rk k∈N in 0, R converging to R and( ) locally complete E a corresponding statement( may) be( proved) using [61, Example 27.27, p. 341-342] where the map T E assigns to each holomorphic function on DR 0 its sequence of Taylor coefficients. Let( us) turn to the space of multipliers for the Schwartz space which is defined by d ∞ d d OM R , E ∶= f ∈C R , E ∀ g ∈ S R ,m ∈ N0, α ∈ A ∶ f g,m,α < ∞ ( ) { ( ) S ( ) Y Y } EXTENSION 33 where β E f g,m,α ∶= sup pα ∂ f x g x Y Y x∈Rd ‰( ) ( )ŽS ( )S d β∈N0 ,SβS≤m (see [71, 30), p. 97]). For simplicity we restrict to the case d = 1. Fix a compactly ∞ 1 supported test function ϕ ∈ Cc R with ϕ x = 1 for x ∈ 0, 4 and ϕ x = 0 for 1 ∞ ( ) ( ) [ ] ( ) x 2 . For f C R, E we set ≥ ∈ ( ∞) ∶ + − − k − − k − + + fj x = f x j Q akϕ 2 x 1 f 2 x 1 j 1 , x ∈ 0, 1 , j ∈ Z, ( ) ( ) k=0 ( ( )) ( ( ) ) [ ] where ∞ 1 + 2j a ∶= , k ∈ N0. k M j − k j=0,j≠k 2 2

Fixing x ∈ 0, 1 , we observe that fj x is well-defined for each j ∈ Z since there are [ ) ( ) k only finitely many summands due to the compact support of ϕ and −2 x−1 → ∞ ∞ ( ) for k → . For x = 1 we have fj 1 = 0 for each j and the convergence of the series ( ) ∞ in E follows from the uniform continuity of f on 0, 1 , f 0 = 0 and ∑k=0 ak = 1 by the case n = 0 in [73, Lemma (iii), p. 625]. For each[ e]′ ∈ E( ′)we note that ∞ ′ ′○ + − − k − ′○ − k − + + e fj x = e f x j Q akϕ 2 x 1 e f 2 x 1 j 1 , x ∈ 0, 1 , j ∈ Z, ( ( )) ( )( ) k=0 ( ( ))( )( ( ) ) [ ] ′ which implies that e ○ fj ∈ E0 by [9, Proposition 3.2, p. 15]. Using the weak-strong principle Corollary 3.29, we obtain that fj ∈ E0 E for all j ∈ Z if E is locally complete. Setting ( ) 1 ρ∶ R → 0, 1 , ρ x ∶= 1 − cos arctan x = 1 − , [ ] ( ) ( ( )) √1 + x2 we deduce from the proof and with the notation of [10, Proposition 2.2, p. 1494] ′ −1 ′ that e ○fj ○ρ = Φ2 ○Φ1 e ○fj is an element of the Schwartz space S R for each ′ ′ e ∈ E . The weak-strong( )( principle) Corollary 3.23 c) yields that fj ○ ρ (∈ S) R, E if E is locally complete. Hence fj ○ ρ ⋅ h2n is Pettis-integrable on R for every( j )∈ Z and n ∈ N0 by [55, 4.8 Proposition,( p.) 15] if E is sequentially complete where

2 ∶ ∶ n −1~2 − d n −x ~2 hn R → R, hn x = 2 n!√π x e , ( ) ( ) ( dx) is the n-th Hermite function. Therefore the Pettis-integral

∶ ○ 2 2 ∶ 2 d Z N0 bn,j f = ⟨fj ρ,h n⟩L = S fj(ρ(x))h n(x) x, j ∈ ,n ∈ , ( ) R is a well-defined element of E if E is sequentially complete. By [10, Theorem 2.1, p. 1496-1497] (cf. [77, Theorem 3, p. 478]) the map K ′ K ∶ ⊗ ∶ 2 Θ OM (R) → s(N)b ̂πs(N), Θ (f) = (bσ(n,j)(f))(n,j)∈N , 2 is an isomorphism where σ∶ N → N0 × Z is the enumeration given by σ(n, j) ∶= (n − 1, (j − 1)~2) if j is odd and σ(n, j) ∶= (n − 1, −j~2) if j is even. Here, we have K ′ ⊗ to interpret Θ (f) as an element of s(N)b ̂πs(N) by identification of isomorphic spaces. Namely, ′ ⊗ ⊗ ′ ′ ′ s(N)b ̂πs(N) ≅ s(N)̂πs(N)b ≅ s(N)εs(N)b ≅ s(N,s(N)b) holds where the first isomorphy is due to the commutativity of ⊗̂π, the second due to the nuclearity of s(N) and the last due to [55, 4.2 Theorem, p. 10] via Ss(N). K ′ Then we intrepret Θ (f) as an element of s(N,s(N)b) by means of j ∈ N z→ a ∈ s N ↦ Q anbσ(n,j) [ ( ) n∈N ] 34 K. KRUSE

(see also (13) below).

6.3. Corollary. If E is a sequentially complete lcHs, then OM R, E ≅OM R εE and the map ( ) ( ) E E ∶ ∶ 2 Θ OM R, E → s N,Lb s N , E , Θ f = bσ(n,j) f (n,j)∈N , ( ) ( ( ( ) )) ( ) ( ( )) E is an isomorphism where we interpret Θ f as an element of s N,Lb s N , E . ( ) ( ( ( ) )) Proof. The spaces OM R and OM R, E are ε-into-compatible by Proposition 2.10 ○ e) as OM R is a Montel( ) space, thus( barrelled,) by [38, Chap. II, §4, n 4, Théorème 16, p. 131].( )

Next, we show that SOM (R) is surjective. We only need to prove that condition (ii) of Theorem 3.14 is fulfilled with FE′ U, E replaced by OM R, E which is [52, 3.16 Condition c)]. Then we may apply( [52,) 3.17 Theorem, p.( 12]) to obtain the ′ ′ surjectivity. Let f ∈ OM R . Using the equicontinuous unconditional Schauder ( ) K basis ψσ(i,j) (i,j)∈N with associated coefficients functionals δi,j ○ Θ = bσ(i,j) given in [10,( Proposition) 3.2, p. 1499] we set ′ ∶ ′ ∶ ′ fn OM R → K, fn f = Q bσ(i,j) f f ψσ(i,j) , ( ) ( ) (i,j)∈N2, S(i,j)S≤n ( ) ( ) ′ for n ∈ N. Along the lines of the proof of Theorem 6.1 b)(ii) we derive that fn ′ ′ ′ ′ 2( ) converges to f in OM R κ. Let f ∈OM R, E . For each e ∈ E and i, j ∈ N we have ( ) ( ) ( ) ○ ΘK ′ ○ ′ ○ ′ ○ d δi,j e f = bσ(i,j) e f = S e f (j−1)~2 ρ x h2(n−1) x x ( ) ( ) R ( ) ( ( )) ( ) ′ ′ E − − d ○ Θ = ⟨e , S f(j 1)~2(ρ(x))h2(i 1)(x) x⟩ = ⟨e ,δi,j (f)⟩ R ′ = e (bσ(i,j)(f)) (12) ○ ⋅ if j is odd since (f(j−1)~2 ρ) h2(i−1) is Pettis-integrable on R. The analogous result holds for even j as well. This implies Rt ′ ′ ′ ′ ○ ′ ○ ′ f (fn)(e ) = fn(e f) = Q bσ(i,j)(e f)f (ψσ(i,j)) (i,j)∈N2, S(i,j)S≤n ′ ′ = e ‰ Q bσ(i,j)(f)f (ψσ(i,j))Ž (i,j)∈N2, S(i,j)S≤n Rt ′ yielding f (fn) ∈ J (E). This shows that [52, 3.16 Condition c)] is fulfilled and the surjectivity of SOM (R) is a consequence of [52, 3.17 Theorem, p. 12] in combination with [52, 3.13 Lemma a), p. 10] as OM (R) is a Montel space. K ○ E ○ 2 Further, we deduce from (12) that (δi,j Θ ,δi,j Θ )(i,j)∈N is a strong family for (OM , E). By [52, 3.17 Theorem, p. 12] the inverse of SOM (R) is given by the −1 ○ Rt R ∶ R map f ↦ J f . Let u ∈OM ( )εE. Then f = SOM (R)(u) ∈OM ( , E) and for 2 each (i, j) ∈ N we get ○ K −1 ○ K −1 Rt ○ K u(δi,j Θ ) = SO (R)(f)(δi,j Θ ) = J ‰ f (δi,j Θ )Ž = bσ(i,j)(f) M (12) ○ E = (δi,j Θ )(SOM (R)(u)) connoting the consistency of our family. 2 2 In order to apply Theorem 6.1 we need spaces ℓV(N ) and ℓV(N , E) of sequences 2 with values in K and E, respectively. In addition, the space ℓV(N ) has to be ′ K∶ ′ 2 isomorphic to s(N,s(N)b) so that Θ OM (R) → s(N,s(N)b) ≅ ℓV(N ) becomes the isomorphism we need for Theorem 6.1. We set 2 2 ∶ N ∀ ∶ ∞ ℓV(N , E) = {x = (xn,j ) ∈ E k ∈ N,B ⊂ s N bounded, α ∈ A x k,B,α < S ( ) Y Y } EXTENSION 35 where E x k,B,α ∶= sup pα T x j, a νk,B j, a Y Y (j,a)∈ωB ( ( )( )) ( ) ∶ × ∶ ∞ ∶ + 2 k~2 with ωB = N B and νk,B ωB → 0, , νk,B j, a = 1 j , plus [ ) ( ) ( ) E ∶ T x j, a = Q anxn,j . ( )( ) n∈N We claim that the map E∶ 2 E ⋅ T ℓV N , E → s N,Lb s N , E , x ↦ T x j, j∈N, (13) ( ) ( ( ( ) )) ( ( )( )) is an isomorphism. First, we remark that for each k ∈ N, bounded B ⊂ s N and α ∈ A we have ( ) E E 2 k~2 T x k,(B,α) = sup sup pα T x j, a 1 + j = x k,B,α S ( )S j∈N a∈B ( ( )( ))( ) Y Y 2 E for all x ∈ ℓV N , E implying that T is an isomorphism into. Let y ∶= yj ∈ s N,Lb s N ,( E . Then) yj ∈ Lb s N , E for j ∈ N and we set xn,j ∶= yj en for( )n ∈ ( ( ( ) )) ( ( ) ) ( ) 2 N where en is the n-th unit sequence in s N . We note that with x ∶= xn,j (n,j)∈N holds ( ) ( ) E T x j, a = Q anxn,j = Q anyj en = yj Q anen = yj a ( )( ) n∈N n∈N ( ) (n∈N ) ( ) for all j ∈ N and a ∶= an ∈ s N since en is a Schauder basis of s N with ( ) ( ) ( ) 2 ( ) associated coefficient functionals a ↦ an. It follows that x ∈ ℓV N , E and the surjectivity of T E. ( ) The next step is to prove that ℓV N2 and ℓV N2, E are ε-into-compatible. Due to Theorem 2.8 we only need to show( ) that T E(,T K )is a consistent generator for ℓV, E . Let u ∈ ℓV N2 εE. Then ( ) ( ) m ( ) m m Q anSℓV(N2) u j, n = Q anu δj,n = u Q anδj,n (14) n=1 ( )( ) n=1 ( ) (n=1 ) for all m ∈ N and a ∶= an ∈ s N . Since m ( ) m ( ) K K ∞ Q anδj,n x = Q anxj,n → T x j, a = T(j,a) x , m → , (n=1 )( ) n=1 ( )( ) ( ) 2 m K 2 ′ for all x ∈ ℓV N , we deduce that ∑n=1 anδj,n m converges to T(j,a) x in ℓV N κ ( ) ( ) 2 ( ) ( ′ ) by the Banach-Steinhaus theorem which is applicable as ℓV N ≅ s N,s N b ≅ OM R is barrelled. We conclude that ( ) ( ( ) ) ( ) m ∞ K E u T = lim u a δ = a S N2 u j, n = T S N2 u j, a (j,a) ∞ Q n j,n Q n ℓV( ) ℓV( ) ( ) m→ (n=1 ) (14) n=1 ( )( ) ( )( ) and thus the consistency of T E,T K . Furthermore, we clearly have( e′ ○)x ∈ ℓV N2 for all x ∈ ℓV N2, E and the map ∶ ′ ⊗ 2 ( ) ( ) Θ OM R → s N b πs N ≅ ℓV N is an isomorphism by [10, Theorem 2.1, p. 1496-1497]( ) and( (13).) ̂ Due( ) to [38,( Chap.) II, §4, n○4, Théorème 16, p. 131] the dual ′ ′ ′ OM R b is an LF-space and thus OM R ≅ OM R b b is the strong dual of an LF-space( ) by reflexivity and therefore webbed( ) ( by Proposition( ) ) 3.12 a). Finally, we can apply Theorem 6.1 (ii) yielding our statement.  6.4. Remark. The actual isomorphism in Corollary 6.3 (without the interpretation) is given by ΘE ∶= T E ○ ΘE with T E from (13) and we have ̃ E E E E K −1 ○ ○ 2 ○ ○ Θ = T Θ = T SℓV(N ) Θ ε idE SO (R) ̃ ( ) M if E is sequentially complete. d d For quasi-complete E the ε-compatibility OM R , E ≅ OM R εE is already contained in [71, Proposition 9, p. 108, Théorème( 1, p. 111].) ( ) 36 K. KRUSE

References [1] A. V. Abanin and L. H. Khoi. Dual of the function algebra A−∞ D and representation of functions in Dirichlet series. Proc. Amer. Math.( ) Soc., 138(10):3623–3635, 2010. [2] A. V. Abanin, L. H. Khoi, and Yu. S. Nalbandyan. Minimal absolutely repre- senting systems of exponentials for A−∞ Ω . J. Approx. Theory, 163(10):1534– 1545, 2011. ( ) [3] A. V. Abanin and V. A. Varziev. Sufficient sets in weighted Fréchet spaces of entire functions. Siberian Math. J., 54(4):575–587, 2013. [4] N. Adasch. Tonnelierte Räume und zwei Sätze von Banach. Math. Ann., 186(3):209–214, 1970. [5] W. Arendt. Vector-valued holomorphic and harmonic functions. Concrete Operators, 3(1):68–76, 2016. [6] W. Arendt and N. Nikolski. Vector-valued holomorphic functions revisited. Math. Z., 234(4):777–805, 2000. [7] W. Arendt and N. Nikolski. Vector-valued holomorphic functions revisited. Math. Z., 252(3):687–689, 2006. [8] D. H. Armitage. Uniqueness theorems for harmonic functions which vanish at lattice points. J. Approx. Theory, 26(3):259–268, 1979. [9] C. Bargetz. Commutativity of the Valdivia-Vogt table of representations of function spaces. Math. Nachr., 287(1):10–22, 2014. [10] C. Bargetz. Explicit representations of spaces of smooth functions and distri- butions. J. Math. Anal. Appl., 424(2):1491–1505, 2015. [11] A. Beurling. The collected works of Arne Beurling, Vol. 2 Harmonic analysis. Contemporary Mathematicians. Birkhäuser, Boston, 1989. [12] K.-D. Bierstedt. Gewichtete Räume stetiger vektorwertiger Funktionen und das injektive Tensorprodukt. PhD thesis, Johannes-Gutenberg Universität Mainz, Mainz, 1971. [13] K.-D. Bierstedt. Gewichtete Räume stetiger vektorwertiger Funktionen und das injektive Tensorprodukt. II. J. Reine Angew. Math., 260:133–146, 1973. [14] K.-D. Bierstedt and S. Holtmanns. Weak holomorphy and other weak proper- ties. Bull. Soc. Roy. Sci. Liège, 72(6):377–381, 2003. [15] R. P. Boas, Jr. Entire functions. Academic Press, New York, 1954. [16] R. P. Boas, Jr. A uniqueness theorem for harmonic functions. J. Approx. Theory, 50(4):425–427, 1972. [17] W. M. Bogdanowicz. Analytic continuation of holomorphic functions with values in a locally convex space. Proc. Amer. Math. Soc., 22(3):660–666, 1969. [18] J. Bonet, P. Domański, and M. Lindström. Weakly compact composition oper- ators on analytic vector-valued function spaces. Ann. Acad. Sci. Fenn. Math., 26:233–248, 2001. [19] J. Bonet, C. Fernández, A. Galbis, and J. M. Ribera. Frames and representing systems in Fréchet spaces and their duals. Banach J. Math. Anal., 11(1):1–20, 2017. [20] J. Bonet, L. Frerick, and E. Jordá. Extension of vector-valued holomorphic and harmonic functions. Studia Math., 183(3):225–248, 2007. [21] J. Bonet, E. Jordá, and M. Maestre. Vector-valued meromorphic functions. Arch. Math. (Basel), 79(5):353–359, 2002. [22] J. B. Cooper. The strict topology and spaces with mixed topologies. Proc. Amer. Math. Soc., 30(3):583–592, 1971. [23] J. B. Cooper. Saks spaces and applications to functional analysis. North- Holland Math. Stud. 28. North-Holland, Amsterdam, 1978. EXTENSION 37

[24] W. F. Donoghue. Distributions and Fourier transforms. Academic Press, New York, 1969. [25] N. Dunford. Uniformity in linear spaces. Trans. Amer. Math. Soc., 44:305–356, 1938. [26] L. Ehrenpreis. Fourier analysis in several complex variables. Pure Appl. Math. (N. Y.) 17. Wiley-Interscience, New York, 1970. [27] J. Fernández, S. Hui, and H. Shapiro. Unimodular functions and uniform boundedness. Publ. Mat., 33(1):139–146, 1989. [28] K. Floret and J. Wloka. Einführung in die Theorie der lokalkonvexen Räume. Lecture Notes in Math. 56. Springer, Berlin, 1968. [29] L. Frerick and E. Jordá. Extension of vector-valued functions. Bull. Belg. Math. Soc. Simon Stevin, 14:499–507, 2007. [30] L. Frerick, E. Jordá, and J. Wengenroth. Extension of bounded vector-valued functions. Math. Nachr., 282(5):690–696, 2009. [31] D. Gilbarg and N. S. Trudinger. Elliptic partial differential equations of second order. Classics Math. Springer, Berlin, 2001. [32] J. Globevnik. Boundaries for polydisc algebras in infinite dimensions. Math. Proc. Cambridge Philos. Soc., 85(2):291–303, 1979. [33] B. Gramsch. Ein Schwach-Stark-Prinzip der Dualitätstheorie lokalkonvexer Räume als Fortsetzungsmethode. Math. Z., 156(3):217–230, 1977. [34] K.-G. Grosse-Erdmann. The Borel-Okada theorem revisited. Habilitation. Fer- nuniv. Hagen, Hagen, 1992. [35] K.-G. Grosse-Erdmann. The locally convex topology on the space of meromor- phic functions. J. Aust. Math. Soc., 59(3):287–303, 1995. [36] K.-G. Grosse-Erdmann. A weak criterion for vector-valued holomorphy. Math. Proc. Camb. Phil. Soc., 136(2):399–411, 2004. [37] A. Grothendieck. Sur certains espaces de fonctions holomorphes. I. J. Reine Angew. Math., 192:35–64, 1953. [38] A. Grothendieck. Produits tensoriels topologiques et espaces nucléaires. Mem. Amer. Math. Soc. 16. AMS, Providence, 4th edition, 1966. [39] W. Gustin. A bilinear integral identity for harmonic functions. Amer. J. Math., 70(1):212–220, 1948. [40] J. Horváth. Topological vector spaces and distributions. Addison-Wesley, Read- ing, Mass., 1966. [41] T. Husain. The open mapping and closed graph theorems in topological vector spaces. Vieweg, Braunschweig, 1965. [42] H. Jarchow. Locally convex spaces. Math. Leitfäden. Teubner, Stuttgart, 1981. [43] E. Jordá. Extension of vector-valued holomorphic and meromorphic functions. Bull. Belg. Math. Soc. Simon Stevin, 12(1):5–21, 2005. [44] E. Jordá. Weighted vector-valued holomorphic functions on Banach spaces. Abstract and Applied Analysis, 2013:1–9, 2013. Article ID 501592. [45] W. Kaballo. Aufbaukurs Funktionalanalysis und Operatortheorie. Springer, Berlin, 2014. [46] M. Yu. Kokurin. Sets of uniqueness for harmonic and analytic functions and inverse problems for wave equations. Math. Notes, 97(3):376–383, 2015. [47] H. Komatsu. Ultradistributions, I, Structure theorems and a characterization. J. Fac. Sci. Univ. Tokyo, Sect. IA, Math. 20:25–105, 1973. [48] H. Komatsu. Ultradistributions, III, Vector valued ultradistributions and the theory of kernels. J. Fac. Sci. Univ. Tokyo, Sect. IA, Math. 29:653–718, 1982. [49] Yu. F. Korobe˘ınik. Inductive and projective topologies. Sufficient sets and representing systems. Math. USSR Izv., 28(3):529–554, 1987. 38 K. KRUSE

[50] G. Köthe. Topological vector spaces II. Grundlehren Math. Wiss. 237. Springer, Berlin, 1979. [51] K. Kruse. Vector-valued Fourier hyperfunctions. PhD thesis, Universität Old- enburg, Oldenburg, 2014. [52] K. Kruse. Weighted vector-valued functions and the ε-product, 2017. arxiv preprint https://arxiv.org/abs/1712.01613. [53] K. Kruse. The approximation property for weighted spaces of differentiable functions, 2018. arxiv preprint https://arxiv.org/abs/1806.02926. [54] K. Kruse. On the nuclearity of spaces of weighted smooth functions, 2018. arxiv preprint https://arxiv.org/abs/1809.06457. [55] K. Kruse. Series representations in spaces of vector-valued functions via Schauder decompositions, 2018. arxiv preprint https://arxiv.org/abs/1806.01889. [56] K. Kruse. Extension of vector-valued functions and weak-strong principles for differentiable functions of finite order, 2019. arxiv preprint. [57] K. Kruse, J. Meichsner, and C. Seifert. Subordination for sequen- tially equicontinuous equibounded C0-semigroups, 2018. arxiv preprint https://arxiv.org/abs/1802.05059. [58] J. Laitila and H.-O. Tylli. Composition operators on vector-valued harmonic functions and Cauchy transforms. Indiana Univ. Math. J., 55(2):719–746, 2006. [59] G. M. Leibowitz. Lectures on complex function algebras. Scott, Foresman and Company, Glenview, Illinois, 1970. [60] R. Meise and B. A. Taylor. Sequence space representations for (FN)-algebras of entire functions modulo closed ideals. Studia Math., 85(3):203–227, 1987. [61] R. Meise and D. Vogt. Introduction to functional analysis. Clarendon Press, Oxford, 1997. [62] S. N. Melikhov. (DFS)-spaces of holomorphic functions invariant under differ- entiation. J. Math. Anal. Appl., 297(2):577–586, 2004. [63] O. Nygaard. Thick sets in Banach spaces and their properties. Quaest. Math., 29(1):59–72, 2006. [64] P. Pérez Carreras and J. Bonet. Barrelled locally convex spaces. Math. Stud. 131. North-Holland, Amsterdam, 1987. [65] M. H. Powell. On K¯omura’s closed-graph theorem. Trans. Am. Math. Soc., 211:391–426, 1975. [66] V. Pták. Completeness and the open mapping theorem. Bull. Soc. Math. France, 86:41–74, 1958. [67] N. V. Rao. Carlson theorem for harmonic functions in Rn. J. Approx. Theory, 12(4):309–314, 1974. [68] L. A. Rubel and A. L. Shields. The space of bounded analytic functions on a region. Ann. Inst. Fourier (Grenoble), 16(1):235–277, 1966. [69] H. H. Schaefer. Topological vector spaces. Grad. Texts in Math. Springer, Berlin, 1971. [70] D. M. Schneider. Sufficient sets for some spaces of entire functions. Trans. Amer. Math. Soc., 197:161–180, 1974. [71] L. Schwartz. Espaces de fonctions différentiables à valeurs vectorielles. J. Analyse Math., 4:88–148, 1955. [72] L. Schwartz. Théorie des distributions à valeurs vectorielles. I. Ann. Inst. Fourier (Grenoble), 7:1–142, 1957. [73] R. T. Seeley. Extension of C∞ functions defined in a half space. Proc. Amer. Math. Soc., 15(4):625–626, 1964. EXTENSION 39

[74] K. Seip. Density theorems for sampling and interpolation in the Bargmann- Fock space I. J. Reine Angew. Math., 429:91–106, 1992. [75] E. L. Stout. The theory of uniform algebras. Bogden and Quigley, Tarrytown- on-Hudson, New York, 1971. [76] M. Valdivia. Sobre el teorema de la gráfica cerrada. Collect. Math., 22(1):51– 72, 1971. [77] M. Valdivia. A representation of the space OM . Math. Z., 177(4):463–478, 1981. [78] A. Wilansky. Modern methods in topological vector spaces. McGraw-Hill, New York, 1978. [79] A. Wiweger. Linear spaces with mixed topology. Studia Math., 20(1):47–68, 1961. [80] D. Zeilberger. Uniqueness theorems for harmonic functions of exponential growth. Proc. Amer. Math. Soc., 61(2):335–340, 1976.

TU Hamburg, Institut für Mathematik, Am Schwarzenberg-Campus 3, Gebäude E, 21073 Hamburg, Germany E-mail address: [email protected]