Mixed topologies on Saks spaces of vector-valued functions

We study Saks spaces of functions with values in a normed space and the associated mixed topologies. We are interested in properties of such Saks spaces and mixed topologies which are relevant for applications in the theory of bi-continuous semigroups. In particular, we are interested if such Saks spaces are complete, semi-Montel, C-sequential or a (strong) Mackey space with respect to the mixed topology. Further, we consider the question whether the mixed and the submixed topology coincide on such Saks spaces and seek for explicit systems of seminorms that generate the mixed topology.


Introduction
This paper is dedicated to Saks spaces of vector-valued functions and their properties.A Saks space is a triple (X, ⋅ , τ ) consisting of a normed space (X, ⋅ ) and a coarser locally convex Hausdorff topology τ on X such that the norm ⋅ is the supremum taken over some directed system of continuous seminorms that generates the τ -topology, see [13].Associated to a Saks space is the mixed topology γ ∶= γ( ⋅ , τ ), which was introduced in [60] and is the finest locally convex Hausdorff, even linear, topology between the ⋅ -topology and τ .Sequentially complete Saks spaces, i.e. (X, γ) is sequentially complete, are needed in the theory of bi-continuous semigroups, which were introduced in [47,48], to treat semigroups on Banach spaces (X, ⋅ ) which are usually not strongly continuous w.r.t. the norm ⋅ but only strongly continuous w.r.t. the coarser topology τ , e.g.dual semigroups, implemented semigroups or transition semigroups like the Ornstein-Uhlenbeck semigroup on the space of bounded continuous functions on a Polish space.
Besides sequential completeness there are several other properties of Saks spaces that are of importance in applications.The Lumer-Phillips generation theorems for bi-continuous semigroups from [45] need knowledge of explicit systems of seminorms that generate the the mixed topology γ because the concept of dissipativity depends on the choice of the system of seminorms.There is another locally convex Hausdorff topology associated to a Saks space, namely the submixed topology γ s ∶= γ s ( ⋅ , τ ), which is defined by an explicit system of seminorms and is in general coarser than γ but has the same convergent sequences as γ, see Definition 2.1.Therefore one is interested in the question when γ and γ s coincide.Moreover, the generation theorems like [45,Theorems 3.10,3.17,Corollary 3.15] need the completeness of the Saks space, i.e. that (X, γ) is complete.A sufficient conditions for γ = γ s is that (X, γ) is a semi-Montel space, which also implies that (X, γ) is a complete semireflexive space and semi-reflexivity is needed for [45,Theorems 3.17] as well.On the other hand, the Lumer-Phillips generation theorem [11,Theorem 3.15,p. 75] for bi-continuous semigroups needs that (X, γ s ) is complete (see [45,Theorem 3.11, Remark 3.12 (b)]).
The question whether γ and γ s coincide is also important for perturbation results of bi-continuous semigroups.If γ = γ s and the Saks space is sequentially complete and C-sequential, i.e. every convex sequentially open subset of (X, γ) is already open, then a bi-continuous semigroup on the corresponding Saks space is already locally, even quasi-, equitight by [42,Theorem 3.17 (b), p. 13].Locally equitight bi-continuous semigroups are sometimes just called "tight" or "local" (see [19,21]) and local equitightness is needed for perturbation theorems like [19,  Apart from its relation to local equitightness its is also known that every bicontinuous semigroup on a sequentially complete C-sequential Saks space is locally, even quasi-, γ-equicontinuous by [35,Theorem 7.4,p. 180] and [42,Theorem 3.17 (a), p. 13].Equicontinuity and local equicontinuity are needed for perturbation results like dissipative perturbations or Desch-Schappacher perturbations [1,26] and the infinitesimal description of Markov processes [25].Sequentially complete C-sequential Saks spaces also play a role in the duality between cost-uniform approximate null-controllability and final state observability, see [44,Theorem 5.18,p. 441].A sufficient condition for (X, γ) being C-sequential is that (X, γ) is a Mackey-Mazur space by [58,Corollary 7.6,p. 52].Here, (X, γ) being a Mackey space means that γ is the Mackey topology of a dual pairing ⟨X, Y ⟩ where Y is a Banach space topologically isomorphic to the strong dual (X, γ) ′ b , and being a Mazur space means that all sequentially γ-continuous linear functionals are already γ-continuous.The question whether (X, γ) is a Mackey space or even a strong Mackey space, i.e. a Mackey space such that σ(Y, X)-compact subsets of Y are γ-equicontinuous, is interesting in itself, see e.g.[49, p. 553] and [35,Propositions 3.4,4.9,p. 161,166].The condition that (X, γ) is a sequentially complete Mackey-Mazur space is also sufficient for the existence of a dual bi-continuous semigroup of a bi-continuous semigroup in the sun dual theory for bi-continuous semigroups, see [43, 3.8 Theorem (b), p. 9 -10].
We are interested in all of the properties listed above in the case of Saks spaces of vector-valued functions.Let us give an outline of our paper.In Section 2 we briefly recall some notions and results from the theory of Saks spaces and give a characterisation of the approximation property of (X, γ) in the case that (X, γ) is a semi-Montel space in Proposition 2. 6.
In Section 3 we start with a Saks space (F (Ω), ⋅ , τ ) of real-or complex-valued functions on a non-empty set Ω such that (F (Ω), ⋅ ) is a Banach space and γ = γ s .We construct a weak E-valued version (F (Ω, E) σ , ⋅ E σ , τ E σ ) of this space in a canonical way, where E is a normed space (or more general a locally convex Hausdorff space), and show in Theorem 3.3 that this triple is a complete Saks space and even complete when equipped with the submixed topology if (F (Ω), ⋅ , τ ) is semi-Montel w.r.t.γ, τ finer than the topology of pointwise convergence and E a Banach space.The proof of this result is based on linearisation via Schwartz' εproduct and as a byproduct we also get a characterisation of (F (Ω), γ) having the approximation property in Corollary 3.4.We apply this result to weak E-valued versions of the Hardy space, weighted Bergman space and the Dirichlet space, whose properties we collect in Corollary 3.5 In Section 4 we consider a different way of defining an E-valued version of (F (Ω), ⋅ , τ ) in Definition 4.2 which is available for some spaces and often stronger in the sense that is a subspace of F (Ω, E) σ and sometimes even a strict subspace, see Proposition 4.4.In Theorem 4.3 we collect some of properties we are interested in of such strong E-valued Saks function spaces.Then we turn to specific examples.Among them are weighted spaces of continuous functions in Corollary 4.5, weighted space of holomorphic functions in Corollary 4.6, weighted kernels of hypoelliptic linear partial differential operators in spaces of smooth functions in Corollary 4.8, in particular weighted spaces of harmonic functions, weighted Bloch spaces in Corollary 4.10, spaces of Lipschitz continuous functions in Corollary 4.11 and spaces of k-times continuously partially differentiable functions on some open bounded set Ω ⊂ R d whose partial derivatives extend continuously to the boundary of Ω and whose partial derivatives of order k are α-Hölder continuous for some 0 < α ≤ 1 in Corollary 4.12.
We close our paper with Section 5 where we characterise the dual of the spaces from the preceding section with respect to a certain submixed topology which sometimes coincides with the mixed topology, see Theorem 5.1.The interest in such a characterisation may also be motivated by the Lumer-Phillips generation theorem [45,Corollary 3.15] which involves the dual w.r.t.mixed topology.

Notions and preliminaries
In this short section we recall some basic notions from the theory of locally convex spaces, Saks spaces and mixed topologies.For a locally convex Hausdorff space X over the field K ∶= R or C we denote by X ′ the topological linear dual space of X.If we want to emphasize the dependency on the locally convex Hausdorff topology τ of X, we write (X, τ ) and (X, τ ) ′ instead of just X and X ′ , respectively.We write (X, τ ) ′ t for the space (X, τ ) ′ equipped with the locally convex topology of uniform convergence on the bounded subsets of (X, τ ) if t = b, and on the absolutely convex compact subsets of (X, τ ) if t = κ.For two locally convex Hausdorff spaces (X, τ ) and (E, τ E ) we denote by L((X, τ ), (E, τ E )) the space of continuous linear maps from (X, τ ) to (E, τ E ).By [54, Chap.I, §1, Définition, p. 18] the ε-product of Schwartz of (X, τ ) and equipped with the topology of uniform convergence on the equicontinuous subsets of (X, τ ) ′ .We identify the tensor product X ⊗ E with the linear finite rank operators in XεE and recap that X has the approximation property of Schwartz if and only if X ⊗ E is dense in XεE for all Banach spaces E (see e.g.[30,Satz 10.17,p. 250]).Moreover, for two locally convex Hausdorff topologies τ 0 and τ 1 on X we write τ 0 ≤ τ 1 if τ 0 is coarser than τ 1 .We write τ E co for the compact-open topology, i.e. the topology of uniform convergence on compact subsets of Ω, on the space C(Ω, E) of continuous functions on a topological Hausdorff space Ω with values in a locally convex Hausdorff space E. If E = K, we just write τ co ∶= τ K co .In addition, we write τ p for the topology of pointwise convergence on the space K Ω of K-valued functions on a set Ω.By a slight abuse of notation we also use the symbols τ E co and τ p for the relative compactopen topology and the relative topology of pointwise convergence on topological subspaces of C(Ω, E) and K Ω , respectively (cf.[40,41,Section 2]).
Let us recall the definition of the mixed topology, [60, Section 2.1], and the notion of a Saks space, [13, I.3.2Definition, p. 27-28], which will be important for the rest of the paper.
2.1.Definition ([42, Definition 2.2, p. 3]).Let (X, ⋅ ) be a normed space and τ a locally convex Hausdorff topology on X such that τ ≤ τ ⋅ where τ ⋅ denotes the topology induced by ⋅ .Then (a) the mixed topology γ ∶= γ( ⋅ , τ ) is the finest linear topology on X that coincides with τ on ⋅ -bounded sets and such that τ ≤ γ ≤ τ ⋅ , (b) the triple (X, ⋅ , τ ) is called a Saks space if there exists a directed system of continuous seminorms Γ τ that generates the topology τ such that In comparison to [42, Definition 2.2, p. 3] we dropped the assumption that the space (X, ⋅ ) should be complete.The mixed topology γ is actually Hausdorff locally convex and the definition given above is equivalent to the one introduced by Wiweger [ It is often useful to have a characterisation of the mixed topology by generating systems of continuous seminorms, e.g. the definition of dissipativity in Lumer-Phillips generation theorems for bi-continuous semigroups from [45] mentioned in the introduction depends on the choice of the generating system of seminorms of the mixed topology.For that purpose we introduce the following auxiliary topology.

2.2.
Definition ([42, Definition 3.9, p. 9]).Let (X, ⋅ , τ ) be a Saks space and Γ τ a directed system of continuous seminorms that generates the topology τ and fulfils (1).We set 0 is the family of all real non-negative null-sequences.For (q n , a n ) n∈N ∈ N we define the seminorm We denote by γ s ∶= γ s ( ⋅ , τ ) the locally convex Hausdorff topology that is generated by the system of seminorms ( ⋅ (qn,an) n∈N ) (qn,an) n∈N ∈N and call it the submixed topology.
(a) We have τ ≤ γ s ≤ γ and γ s has the same convergent sequences as γ.(b) If (i) for every x ∈ X, ε > 0 and q ∈ Γ τ there are y, z ∈ X such that x = y + z, q(z) = 0 and y ≤ q(x) The submixed topology γ s was originally introduced in [60, Theorem 3.1.1,p. 62] where a proof of Remark 2.3 (b) can be found, too.The following notions will also be needed, which were introduced in [13 ) is a Banach space by [13, I.1.18Proposition, p. 15].
We close this section with the following observation concerning the approximation property of (X, γ) in the semi-Montel case, whose proof is an adaptation of [29, Theorem 4.6 (i)⇔(ii), p. 651-652] where X = Lip 0 (Ω) is the space of K-valued Lipschitz continuous functions on a metric space Ω that vanish at the origin (see Corollary 4.11).

Saks spaces of weak vector-valued functions
In this section we introduce Saks spaces of weak vector-valued functions.We use a linearisation based on the ε-product to show that they are complete w.r.t. the mixed and the submixed topology if their scalar-valued version is semi-Montel, τ p ≤ τ and they have values in a Banach space.
Let (F (Ω), ⋅ ) be a Banach space of K-valued functions on a non-empty set Ω such that τ p ≤ τ ⋅ .We recall from [40, p. 31] a canonical construction of a weak vector-valued version of such a space.For a locally convex Hausdorff space E over K with directed system of seminorms Γ E generating its topology we define the space of weak E-valued F -functions by ) is a normed space where ⋅ E σ ∶= ⋅ σ, ⋅ E .Now, let τ be an additional locally convex Hausdorff topology on F (Ω) such that (F (Ω), ⋅ , τ ) is a Saks space and γ ∶= γ( ⋅ , τ ) = γ s ( ⋅ , τ ).Then, by Definition 2.2, a directed system of seminorms that generates γ is given by for (q n ) n∈N ⊂ Γ τ , where Γ τ is a directed system of continuous seminorms that generates the topology τ and fulfils (1), and (a n ) n∈N ∈ c + 0 .We set f σ,(qn,an) n∈N ,p ∶= sup for all f ∈ F (Ω, E) σ .So the system of seminorms ( f σ,(qn,an) n∈N ,p ) (qn,an) n∈N ∈N ,p∈Γ E induces a locally convex Hausdorff topology on F (Ω, E) σ which we denote by γ E σ,s .
3.1.Remark.Let (F (Ω), ⋅ , τ ) be a Saks space of K-valued functions on a nonempty set Ω such that (F (Ω), ⋅ ) is a Banach space, τ p ≤ τ ⋅ and γ( ⋅ , τ ) = γ s ( ⋅ , τ ), and E a locally convex Hausdorff space over K with directed system of seminorms Γ E generating its topology.Then the topology γ E σ,s does not depend on the choice of the system of seminorms that generates γ ∶= γ( ⋅ , τ ).Indeed, let Γ γ be a another system of seminorms that generates γ.Then for every (q n ) n∈N ⊂ Γ τ , Γ τ as above, and (a n ) n∈N ∈ c + 0 there are C 0 ≥ 0 and r 0 ∈ Γ γ such that f σ,(qn,an) n∈N ≤ C 0 r 0 (f ) for all f ∈ F (Ω). On the other hand, for every and r 1 ∈ Γ γ there are for all f ∈ F (Ω, E) σ and p ∈ Γ E .Thus the system of seminorms given by ) is a Saks space where τ E σ is the locally convex Hausdorff topology on F (Ω, E) σ generated by the system of seminorms given by for q ∈ Γ τ and Γ τ as above.Indeed, this follows from Definition 2.1 and the observation 2 and the definitions of τ E σ and γ E σ,s .For a linear space F (Ω) of K-valued functions on a non-empty set Ω and x ∈ Ω we define the linear functional ) is a complete Saks space.Proof.First, we show that χ is well-defined and linear.Since γ for all x ∈ Ω (cf.[41, 4.2 Remark, p. 12]).We note that where τ E denotes the locally convex Hausdorff topology of ′ and the semireflexivity of the semi-Montel space F (Ω) γ , we note that there is Thus the map χ is well-defined and it is easily seen to be linear as well.
Second, we show that χ is injective and continuous.Let Γ τ be a directed system of continuous seminorms that generates the topology τ and fulfils (1).For (q n ) n∈N ⊂ Γ τ and (a n ) n∈N ∈ c + 0 we define U qnan ∶= {f ∈ F (Ω) q n (f )a n < 1} and note that the sets form a base of γ-neighbourhoods of zero by Definition 2.2 since γ = γ s by condition (ii) of Remark 2.3 (b) and Remark 2.5.By the bipolar theorem we have where acx(W (qn,an) n∈N ) denotes the closure in (F (Ω), γ) ′ κ of the absolutely convex hull acx(W (qn,an) n∈N ) of W (qn,an) n∈N ∶= ⋃ n∈N U ○ qnan (see [28,8.2.4 Corollary,p. 149]).Due to [28, 8.4, p. 152, 8.5, p. 156-157] the topology of F (Ω) γ εE is generated by the seminorms where Γ E denotes a system of seminorms that generates τ E .By the continuity of u ∈ F (Ω) γ εE we have On the other hand, for y ∈ acx(W (qn,an) n∈N ) there are m ∈ N, and we deduce By the first part of the proof there is and for all u ∈ F (Ω) γ εE.Hence χ is injective and continuous.Third, we show that χ is surjective and note that (4) implies that the inverse of χ is continuous.Due to Remark 2.5 (d) we have . Hence the surjectivity of χ is a consequence of [40, 5.5   if Ω is discrete, the weighted space F (Ω, E) = Hv(Ω, E) of holomorphic functions from Corollary 4.6 and the weighted kernel F (Ω, E) = C P v(Ω, E) of a hypoelliptic linear partial differential operator from Corollary 4.8 even for quasi-complete locally convex Hausdorff spaces E. However, the proof is different.Theorem 3.3 also allows us to characterise (F (Ω), γ) having the approximation property by approximation in (F (Ω, E) σ , γ E σ,s ).3.4.Corollary.Let (F (Ω), ⋅ , τ ) be a semi-Montel Saks space of K-valued functions on a non-empty set Ω such that τ p ≤ τ .Then the following assertions are equivalent.
) for every complete locally convex Hausdorff space E over K.  3), (4), p. 205] if Ω is discrete.We close this section with an application of Theorem 3.3 to some spaces of integrable holomorphic functions.We denote by H(D) the space of C-valued holomorphic functions on D. For 1 ≤ p < ∞ the Hardy space is defined by and the weighted Bergman space for α > −1 by The Dirichlet space is defined by then τ co on H p is generated by the directed system of continuous seminorms ( ⋅ p,s ) 0<s<1 given by α , then τ co on A p α is generated by the directed system of continuous seminorms ( ⋅ α,p,r ) 0<r<1 given by then τ co on D is generated by the directed system of continuous seminorms ( ⋅ D,r ) 0<r<1 given by Proof.(c) In all the cases we note that the τ co -compactness of B ⋅ is obtained from [17, p. 4-5] (which uses that B ⋅ is compact in the Montel space (H(D), τ co ) since B ⋅ is relatively compact there and its closedness is a consequence of Fatou's lemma).It follows that (F (D), ⋅ , τ co ) is a semi-Montel Saks space by Remark 2.5 (a).Further, condition (ii) of Remark 2.3 (b) yields that γ( ⋅ , τ co ) = γ s ( ⋅ , τ co ).
(a)+(b) Due to part (c) and τ p ≤ τ co we have that ) is a Saks space, which is complete if E is a Banach space, by Remark 3.2 and Theorem 3.3.Since the countable system of seminorms (d) For 0 < s < 1 we have p as well as f p = sup 0<s<1 f p,s for all f ∈ H p .Furthermore, for 0 < s < r < 1 we remark that p for all f ∈ A p α and 0 < r < 1.Now, for 0 < r < 1 we choose 0 < s < 1 − r.We deduce from the mean value equality for holomorphic functions and Hölder's inequality that Thus τ co on A p α is generated by the directed system of continuous seminorms ( ⋅ α,p,r ) 0<r<1 .
(f) We observe that f D = sup 0<r<1 f D,r for all f ∈ D.Moreover, we obtain by Hölder's inequality that for all z ∈ D r and f ∈ D. Now, for 0 < r < 1 we choose 0 < s < 1 − r.From Cauchy's inequality we deduce the estimate for all f ∈ D, implying that τ co on D is generated by the directed system of continuous seminorms ( ⋅ D,r ) 0<r<1 .

Saks spaces of vector-valued functions
This section is dedicated to Saks spaces F (Ω, E) of vector-valued functions which are often stronger than the spaces F (Ω, E) σ of weak vector-valued functions from the preceding section (see Proposition 4.4).In order to derive certain systems of seminorms on such spaces which generate the mixed topology we need to recall some results for completely regular Hausdorff spaces Ω (see [27,  Let Ω be a completely regular Hausdorff space, (K n ) n∈N a strictly increasing sequence of compact subsets of Ω and (a n ) n∈N a strictly decreasing positive null-sequence.Then there is w ∈ W + usc,0 (Ω) such that supp w ⊂ ⋃ n∈N K n and w(x) = a 1 for x ∈ K 1 and a n+1 ≤ w(x) ≤ a n for x ∈ K n+1 ∖ K n and n ∈ N. If Ω is locally compact and K n ⊂ Kn+1 for every n ∈ N, then we may choose w ∈ C + 0 (Ω).Here, supp w denotes the support of w and Kn+1 the set of inner points of K n+1 .

Definition.
Let Ω and Λ be non-empty sets, v∶ linear map where E Λ denotes the space of functions from Λ to E. We define the space For a non-empty set Λ we denote by N Λ the familiy of finite subsets of Λ.If Λ is a topological space, we denote by K Λ the family of compact subsets of Λ.

Theorem.
Let Ω and Λ be non-empty sets, v∶ Λ → (0, ∞), (E, ⋅ E ) a normed space over K, G(Ω, E) a linear subspace of S be a family of subsets of Λ such that S is closed under finite unions, Λ = ⋃ S∈S S and denote by τ E S the locally convex Hausdorff topology generated by the directed system of seminorms for S ∈ S. Then the following assertions hold.
is generated by the system of seminorms where where (x n ) n∈N is a sequence in Λ and (a n ) n∈N ∈ c + 0 .(e) Let Λ be a completely regular Hausdorff space and set is generated by the system of seminorms for w ∈ W 0 .If Λ is locally compact, we may replace W 0 by C + 0 (Λ).Proof.(a) First, we note that the system of seminorms (q E S ) S∈S is directed and Hausdorff since S is closed under finite unions, ⋅ E a norm by assumption and  bn) n∈N for all f ∈ F v(Ω, E).On the other hand, f E ({zn},an) n∈N = f E (zn,an) n∈N for all f ∈ F v(Ω, E) and every sequence (z n ) n∈N in Λ.Thus statement (d) follows from part (c).
(e) We denote by ω E b and ω E usc the locally convex Hausdorff topologies generated by ( ⋅ E w ) w∈W + b,0 (Λ) and ( ⋅ E w ) w∈W + usc,0 (Λ) , respectively.First, we prove that the identity map id∶ and so the continuity of id.
Second, we prove that id∶ Ω, E), which yields the continuity of id by part (c) and the assumption Then the following assertions hold.E).Suppose for (b)-(d) that S is a family of subsets of Λ such that S is closed under finite unions and Λ = ⋃ S∈S S, and is generated by the system of seminorms ( ⋅ E w ) w∈W0 .If Λ is locally compact, we may replace W 0 by C + 0 (Λ).Proof.(a) Due to the first part of condition (i) we obtain that F v(Ω, E) is a linear subspace of F v(Ω, E) σ .The second part of condition (i) implies that for all f ∈ F v(Ω, E).Together with condition (ii) this yields that (b) Let (S n ) n∈N be a sequence in S and (a n ) n∈N ∈ c + 0 .We have by part (a) Using the second part of condition (i) we get as in part (a) that for all f ∈ F v(Ω, E).In combination with condition (ii) we obtain the estimates ⋅ σ,(Sn,an for w ∈ W 0 .We deduce that the system of seminorms given by for w ∈ W 0 generates the topology γ s ( ⋅ E σ , τ E KΛ,σ ) on F v(Ω, E) σ by Remark 3.1 and Remark 3.2.Similar to the proofs of part (a) and (b) we obtain that ⋅ σ,w, ⋅ E ≤ ⋅ E w ≤ 2 ⋅ σ,w, ⋅ E on F v(Ω, E) for every w ∈ W 0 , yielding that γ s ( ⋅ E , τ E KΛ ) is generated by the system of seminorms ( ⋅ E w ) w∈W0 by part (b).Condition (i) of Proposition 4.4 means that the tuple (T E , T ) is strong for (F v, E) in the sense of [38, Definition 2.2 (b), p. 4].In our first example of this section we consider weighted Saks spaces of continuous vector-valued functions on a completely regular Hausdorff space and a sufficient condition for their completeness involves the notion of a k R -space.A completely regular space Ω is called a k R -space if for any completely regular space Y and any map f ∶ Ω → Y , whose restriction to each compact K ⊂ Ω is continuous, the map is already continuous on Ω (see [8, (2.3.7)Proposition, p. 22]).Moreover, a topological space Ω is called a k-space if it fulfils the following condition: Examples of Hausdorff k R -spaces are completely regular Hausdorff k-spaces by [18,3.3.21 Theorem,p. 152].In particular, metrisable spaces and DFM-spaces, i.e. strong duals of Fréchet-Montel spaces, are completely regular Hausdorff k-spaces by [18,3.3.20 Theorem,p. 152] and [36,4.11 Theorem (5), p. 39], respectively.For a non-empty completely regular Hausdorff space Ω, a continuous function v∶ Ω → (0, ∞) and a normed space (E, ⋅ E ) over K we set where C(Ω, E) is the space E-valued continuous functions on Ω.Further, we define Cv(Ω) ∶= Cv(Ω, K) and ⋅ ∶= ⋅ K .Setting Λ ∶= Ω, G(Ω, E) ∶= C(Ω, E), q E ∶= 0 and T E (f ) ∶= f for f ∈ C(Ω, E), we note that F v(Ω, E) = Cv(Ω, E), τ E KΩ = τ E co on Cv(Ω, E) and conditions (i) and (ii) of Proposition 4.4 are fulfilled.

Corollary.
Let Ω be a non-empty completely regular Hausdorff space, v∶ Ω → (0, ∞) continuous and (E, ⋅ E ) a normed space over K. Then the following assertions hold.
and the mixed topology γ( ⋅ E , τ E co ) is generated by the system of seminorms where is also generated by the system of seminorms where is also generated by the system of seminorms for w ∈ W 0 .If Ω is locally compact, we may replace W 0 by C + 0 (Ω).Proof.Due Theorem 4.3 and our observations above we only need to prove that γ( ⋅ E , τ E co ) = γ s ( ⋅ E , τ E co ) from part (a), and parts (b), (c) and (e).(a) We show that condition (i) of Remark 2.3 (b) is fulfilled.We only need to adjust the proof of [60, Example D), p. 65-66] to the weighted vector-valued case, which we do for the sake of the reader.Let f ∈ Cv(Ω, E), ε > 0 and K ⊂ Ω be compact.Since The set Ω ∖ G is closed and disjoint with the compact set K ⊂ Ω.By [8, (2.1.5)Proposition, p. 17] the complete regularity of Ω implies that there is a continuous function u∶ Ω → [0, 1] such that u K = 0 and u Ω∖G = 1.Now, we set g ∶= (1 − u)f and h ∶= uf and note that f = g + h and g, h ∈ Cv(Ω, E).Due to the properties of u we have q E K (h) = 0 and (c) Let C b (Ω, E) ∶= Cṽ(Ω, E) for ṽ(x) ∶= 1, x ∈ Ω, and set ⋅ E ∞ ∶= ⋅ Cṽ(Ω,E) .By part (f) the multiplication operator [13, p. 112] and note that k R -spaces are exactly the Kcomplete spaces by [13, p. 80]) and (E, ⋅ E , τ ⋅ E ) a complete Saks space as ) is a Mackey space, which is strong if E is a Banach space, by part (f) and [33,Theorem 3.4,p. 165].Let Ω be a completely metrisable space.Then ) is a Mackey space, which is strong if E is a Banach space, by part (f), [32,Theorem 2,p. 35] and [31,Theorem 3.7,p. 202].Using the topological isomorphism M E v from part (c), we note that both statements remain valid if we replace C b (Ω, E) by Cv(Ω, E) and ⋅ E ∞ by ⋅ E .We remark that (Cv(Ω), ⋅ , τ co ) is semi-Montel by [41, 3.9 Example (i), p. 10] if Ω is discrete.In the case C b (Ω, E) the statement from Corollary 4.5 (a) that ) is a Mazur space if Ω is Polish and E a Banach space, i.e. that every sequentially γ( ⋅ E , τ E co )-continuous linear functional on Cv(Ω, E) is already γ( ⋅ E , τ E co )-continuous.Next, we consider subspaces of Cv(Ω, E).Let (E, ⋅ E ) be a normed space over C. For a non-empty open subset Ω of a complex locally convex Hausdorff space let H(Ω, E) be the space of holomorphic functions f ∶ Ω → E, i.e. the space of Gâteauxholomorphic and continuous functions f ∶ Ω → E (see [15, Definition 3.6, p. 152]), and for a continuous function v∶ Ω → (0, ∞) we set Further, we define Hv(Ω) ∶= Hv(Ω, C) and   4.7.Remark.Let (E, ⋅ ) be a Banach space over C and 1 ≤ p < ∞.We may also define a strong E-valued version of the Hardy H p from Corollary 3.5.Let However, in contrast to the case p = ∞, we only have the strict inclusion Let us turn to another subspace of Cv(Ω, E).For a non-empty open set Ω ⊂ R d and a normed space (E, ⋅ E ) over K we denote by C ∞ (Ω, E) the space of infinitely continuously partially differentiable E-valued functions on Ω and by For K = C and a polynomial P on R d with constant complex coefficients we define the linear partial differential operator P (∂) E ∶= P ((∂) E ) on C ∞ (Ω, E) in the usual way and its kernel For a continuous function v∶ Ω → (0, ∞) we define the weighted kernel where for all f ∈ Bv(D, E) where the integral in the estimate above is a Bochner integral (cf.[38, Corollary 3.8, p. 9-10] for the case E = C), and for every 0 < s < r < 1 Further, we define Lip 0 (Ω) ∶= Lip 0 (Ω, K) and ⋅ ∶= ⋅ K .Setting Λ ∶= Ω wd ∶= {(x, y) ∈ Ω 2 x ≠ y}, v∶ Λ → (0, ∞), v(x, y) ∶= 1 d(x,y) , G(Ω, E) ∶= {f ∶ Ω → E f (0) = 0}, q E ∶= 0 and T E (f )(x, y) ∶= f (x) − f (y) for (x, y) ∈ Λ and f ∈ G(Ω, E), we observe that F v(Ω, E) = Lip 0 (Ω, E) and conditions (i) and (ii) of Proposition 4.4 are fulfilled.4.11.Corollary.Let (Ω, d) be a pointed metric space and (E, ⋅ E ) a normed space over K. Then the following assertions hold. (a) for S ∈ {N Ω wd , K Ω wd }.
Proof.(a) First, we note that for compact K ⊂ Ω wd the projections π 1 (K) and π 2 (K) on the first and second component, respectively, are compact in Ω and ≤ τ E co on Lip 0 (Ω, E).On the other hand, for every ε > 0 and compact K ⊂ Ω we have with where Ω is compact and W 0 = C + 0 (Ω wd ).For a normed space (E, ⋅ E ) over K and k ∈ N 0 we denote by C k (Ω, E) the space of k-times continuously partially differentiable E-valued functions on a non-empty open bounded set Ω ⊂ R d .We define the space of k-times continuously partially differentiable E-valued functions on Ω whose partial derivatives up to order k are continuously extendable to the boundary of Ω by β ≤ k} which we equip with the norm given by where we use the same symbol for the unique continuous extension of (∂ β ) E f to Ω.The space of functions in C k (Ω, E) such that all its k-th partial derivatives are α-Hölder continuous with 0 < α ≤ 1 is given by as well as ⋅ ∶= ⋅ K .Let E be Banach space.Then for every β ∈ N d 0 with β = k and f ∈ C k,α (Ω, E) the unique continuous extension of the partial derivative (∂ β ) E f to Ω is α-Hölder continuous and the extension has the same Hölder constant by [57, and (E, ⋅ E ) a Banach space over K. Then the following assertions hold.
and note that for compact K ⊂ Ω wd the projections π 1 (K) and π 2 (K) on the first and second component, respectively, are compact in Ω and . On the other hand, fix some y 0 ∈ Ω.For every ε > 0 we have with ) is a Saks space.Furthermore, we deduce that ) by the definition of the mixed topology.Since ) is C-sequential by [ for w ∈ W 0 (without the modification the weights w depend on β as well).
5. The dual space of (F v(Ω, E), γ s ( ⋅ E , τ E NΛ )) In our closing section we give a characterisation of the dual space of the space F v(Ω, E) from Definition 4.2 w.r.t. the submixed topology γ s ( ⋅ E , τ E NΛ ).We know from the preceding section that this submixed topology often coincides with the mixed topology, at least if Ω is discrete or E = K.Our proof is an adaptation of the proof of the corresponding result [29, Theorem 5.1, p. 652] for the case F v(Ω, E) = Lip 0 (Ω, E).For a normed space (E, ⋅ E ) we denote by E ⊕ 1 E the space E × E equipped with the norm ⋅ E⊕1E given by (x, y) E⊕1E ∶= x E + y E for x, y ∈ E. By ℓ 1 (N, (E ⊕ 1 E) * ) we denote the space of (E ⊕ 1 E) * -valued sequences y = (y n ) n∈N such that y 1 ∶= ∑ ∞ n=1 y n (E⊕1E) * < ∞.
E) by Remark 3.2 and Theorem 4.3 (b).(c) The completeness of (F v(Ω, E), γ s ( ⋅ E , τ E S )) follows from part (b), Remark 3.2 and Theorem 3.3.The completeness of ( 5 (a) and condition (ii) of Remark 2.3 (b).Due to Theorem 4.3 (d) γ( ⋅ , τ KΛ ) is generated by the system of seminorms every subset of Ω is open, and a subset of Ω is compact if and only if it is finite.Thus τ E co = τ E KΩ = τ E NΩ and statement (b) follows from part (a), Theorem 4.3 (d) and Mackey's theorem.

4 . 6 .
Hv(Ω, E) and conditions (i) and (ii) of Proposition 4.4 are fulfilled.Corollary.Let Ω be a non-empty open subset of a locally convex Hausdorff space X over C, v∶ Ω → (0, ∞) continuous and (E, ⋅ E ) a normed space over C. Then the following assertions hold.

For a normed
space (E, ⋅ E ) over C and a continuous function v∶ D → (0, ∞) with D = {z ∈ C z < 1} we define the Bloch type space Bv
Remark.Let (X, ⋅ ) be a normed space and τ a locally convex Hausdorff topology on X. Set X * ∶= (X, ⋅ ) ′ and denote by ⋅ X * the dual norm on X * .
[30,th, since F (Ω) γ and E are complete, F (Ω) γ εE is also complete by[30, Satz  10.3, p. 234], implying the completeness of Hence τ E S is a locally convex Hausdorff topology with τ E (d) Let (a n ) n∈N ∈ c + 0 and (S n ) n∈N be a sequence of finite subsets of Λ with cardinality m n ∶= S n for n ∈ N. Then every S n , n ∈ N, is of the form S n = {s n 1 , . . ., s n mn } with distinct elements s S ≤ τ ⋅ E and (F v(Ω, E), ⋅ E , τ E S ) a Saks space.(b)Thecountablesystem of seminorms (q Sn ) n∈N generates τ E S by our assumption on S. Hence τ E S is metrisable and so (F v(Ω, E), γ( ⋅ E , τ E S )) C-sequential by[46,  Proposition 5.7, p. 2681[46,  Proposition 5.7, p.  -2682]].(c)This follows from part (a) and the definition of γ s ( ⋅ E , τ E S ).