Weighted Vector-Valued Holomorphic Functions on Banach Spaces

Abstract

We study the weighted Banach spaces of vector-valued holomorphic functions defined on an open and connected subset of a Banach space. We use linearization results on these spaces to get conditions which ensure that a function 𝑓 defined in a subset 𝐴 of an open and connected subset 𝑈 of a Banach space 𝑋, with values in another Banach space 𝐸, and admitting certain weak extensions in a Banach space of holomorphic functions can be holomorphically extended in the corresponding Banach space of vector-valued functions.

1. Introduction, Notation, and Preliminaries

Let 𝐸 be a locally convex space. The problem of deciding when a function 𝑓∶Ω⊂ℂ→𝐸 is holomorphic whenever 𝑢∘𝑓∈𝐻(Ω) for each 𝑢∈𝐸^′ goes back to Dunford [1], who proved that this happens when 𝐸 is a Banach space. Grothendieck [2] extended the result for 𝐸 being quasicomplete. Bogdanowicz [3] gives extension results through weak extension, that is, he proved between other results that if Ω₁⊂Ω₂⊂ℂ are two domains (open and connected subsets), 𝐸 is a complex, sequentially complete, and locally convex Hausdorff space, and 𝑓∶Ω₁→𝐸 satisfies that 𝑢∘𝑓 admits holomorphic extension for each 𝑢∈𝐸^′, then 𝑓 admits a holomorphic extension to Ω₂. More recently Grosse-Erdmann, Arendt and Nikolski, Bonet, Frerick, Wengenroth, and the author have given results in this way smoothing the conditions on Ω₁ and also requiring extensions of 𝑢∘𝑓 only for a proper subset 𝐻⊆𝐸^′ (cf. [4–8]). Also, Laitila and Tylli have recently discussed the difference between strong and weak definitions for important spaces of vector-valued functions [9, Section 6].

Our main aim here is to analyze a weak criterion for holomorphy and to give extension results for the Banach spaces of holomorphic functions defined on a nonvoid open subset 𝑈 of a Banach space 𝑋. To obtain these extension results, we use linearization results, that is, theorems which permit to identify classes of vector valued functions defined in 𝑈 and with values in 𝐸 with continuous linear mappings from a certain space 𝐺 and with values in 𝐸. Recent work of Beltrán [10], Carando and Zalduendo [11], and Mujica [12] is devoted to get linearization results. We use for our extension results also linearization results obtained by Bierstedt in [13, 14].

Our notation for the Banach spaces, locally convex spaces, and functional analysis is standard. We refer the reader to [15–17]. For a locally convex space 𝐹 which is nonnormed, we denote by 𝐹^′ its topological dual. For a Banach space (𝐸,‖⋅‖), the dual of 𝐸 is denoted by 𝐸^∗. We mainly deal with Banach spaces. The absolutely convex hull of a subset 𝐶 of 𝐸 is denoted by Γ(𝐶), and the closure of 𝐶 is denoted by 𝐶. If the closure is taken with respect to other topology 𝜏, it will be denoted by 𝐶^𝜏. (𝐸,𝑤) and (𝐸^∗,𝑤∗) are 𝐸 and 𝐸^∗ endowed with the weak (𝜎(𝐸,𝐸^∗)) and the weak* (𝜎(𝐸^∗,𝐸)) topology, respectively. The open unit ball of 𝐸 will be denoted by 𝐵_𝐸. A subset 𝑀⊆𝐸 (𝑀⊆𝐸^∗) is said to be total if span(𝑀) is (𝜎(𝐸^∗,𝐸)) dense. By the Hahn Banach theorem, 𝑀 being total in 𝐸^∗ is equivalent to 𝑀 being separating, that is, if 𝑒∈𝐸 and 𝑢(𝑒)=0 for all 𝑢∈𝑀, then 𝑒=0. 𝑀⊆𝐸^∗ is said to be norming if 𝑀 is bounded, and its associated functional 𝑞_𝑀∶𝐸→ℝ, 𝑒↦sup{|𝑢(𝑒)|∶𝑢∈𝑀} defines an equivalent norm in 𝐸, that is, if the polar set 𝑀^∘∶={𝑒∈𝐸∶|𝑢(𝑒)|≤1forall𝑢∈𝑀} defines an equivalent (closed) unit ball in 𝐸. It is immediate that if 𝑀 is norming then it is also separating. If 𝑞_𝑀 is the norm of 𝐸, then 𝑀 is called 1-norming. A subset 𝑀⊂𝐸 is called norming or total when we consider 𝐸⊂𝐸^∗∗, that is, 𝑞_𝑀 defines an equivalent norm in 𝐸^∗. A subspace 𝐻 of 𝐸^∗ is said to determine boundedness whenever all the 𝜎(𝐸,𝐻)-bounded subsets of 𝐸 are (𝜎(𝐸,𝐸^∗)-) bounded. A subspace 𝐻⊆𝐸^∗ is said to be norming if 𝐵_𝐻 is a norming subset of 𝐸. We give below a relation between these concepts. The result is given in [6, Proposition 7, Remark 8] in the more general context of the Fréchet spaces, though in this paper the norming subspaces are called almost norming. We give a proof for the Banach case because it is very transparent.

Proposition 1 (see [6]). Let 𝐸 be a Banach space. A subspace 𝐻⊂𝐸^∗ is norming if and only if 𝐻 determines boundedness in 𝐸.

Proof. It is standard to check that 𝑞_𝐵𝐻=𝑞_𝐵𝐻 on 𝐸. From this, it follows that 𝐻 is norming if and only if 𝐻 is. Assume first that 𝐻 is norming. This implies that 𝐻 is separating on 𝐸, and then, we can consider the algebraic inclusion 𝐸↪𝐻^∗. By the very definition, 𝑞_𝐵𝐻 is the restriction of ‖⋅‖_𝐻^∗ to 𝐸. The hypothesis 𝐻 norming means that 𝐸 is isomorphic to (𝐸,𝑞_𝐵𝐻), which is a subspace of (𝐻^∗,‖⋅‖_𝐻^∗). By the Uniform Boundedness Principle, the 𝜎(𝐸,𝐻) bounded subsets of 𝐸 are norm bounded in 𝐻^∗, and then, ‖⋅‖ bounded since the norms are supposed to be equivalent in 𝐸. Conversely, let one assume that 𝐻 determines boundedness in 𝐸. This implies, again by the Uniform Boundedness Principle, that the identity 𝐼∶(𝐸,‖⋅‖_𝐻^∗)→(𝐸,‖⋅‖) is bounded. Hence, there exists 𝐶≥1 such that ‖𝑒‖≤𝐶‖𝑒‖_𝐻^∗=𝐶𝑞_𝐵𝐻(𝑒) for each 𝑒∈𝐸, which implies that 𝐻 is norming.

Thus, the property of being norming for subspaces in 𝐸^∗ is between weak*-dense and strongly dense.

Let 𝑀=(𝑚_𝑖)_𝑖∈𝐼 be a bounded subset of 𝐸 and 𝐼 an index set. Let 𝑙₁(𝑀)∶=𝑥∈𝐸∶exists𝑎_{𝑖𝑖∈𝐼}∈𝑙₁(𝐼)suchthat𝑥=𝑎_𝑖𝑚_𝑖, (1) equipped with the norm which makes it isomorphic to a quotient of 𝑙₁(𝐼). We will use the following lemma, which we supposed to be well known.

Lemma 2. Let 𝑀⊆𝐵_𝐸 be a norming subset. Then, the injection of 𝑙₁(𝑀) in 𝐸 is an onto isomorphism.

Proof. The hypothesis on 𝑀 yields that there exist 𝑐,𝐶>0 such that 𝑐𝐵_𝐸^∗⊆𝑀^∘⊆𝐶𝐵_𝐸^∗. (2) Hence, we take polars and apply the bipolar theorem to get 1𝐶𝐵_𝐸⊆Γ(𝑀)⊆1𝑐𝐵_𝐸. (3) Let 𝐶_𝐸 be the equivalent open unit ball in 𝐸 such that 𝐶_𝐸=Γ(𝑀). We define 𝑇∶𝑙₁(𝐼)→𝐸, (𝑎_𝑖)_𝑖↦∑_𝑖∈𝐼𝑎_𝑖𝑚_𝑖. 𝑇 is clearly bounded. Moreover, Γ(𝑀)⊆𝑇(𝐵_𝑙1(𝐼)), and then, 𝐶_𝐸⊆Γ(𝑀)⊆𝑇𝐵_𝑙1(𝐼). (4) We get from the Schauder lemma [16, Lemma 3.9] that 𝑇 is open and then surjective. We conclude from the very definition of 𝑙₁(𝑀).

Remark 3. If we assume in Lemma 2 that 𝑀 is 1-norming, then the isomorphism is an isometry.

We see below that if the bounded subset 𝑀 is not norming, then the assertion is not true in general.

Remark 4 (Bonet). The assertion in Lemma 2 implies that if 𝑀⊆𝐸 is bounded, then 𝑥∈Γ(𝑀) if and only if for each 𝜀>0 there exist sequences (𝛼_𝑛)_𝑛∈(1+𝜀)𝐵_𝑙1 and (𝑚_𝑛)_𝑛⊂𝑀 such that 𝑥=∑_𝑛𝛼_𝑛𝑚_𝑛. This is not in general true if we assume that 𝑀 is only to be bounded. Valdivia showed (see [17, Example 3.2.21]) that in every infinite dimensional Banach space there is an absolutely convex bounded subset 𝐵⊂𝐸 which is not closed such that 𝐸_𝐵∶=(span(𝐵),𝑝_𝐵) is a Banach space with closed unit ball 𝐵, where 𝑝_𝐵(𝑥)∶=inf{𝜆>0,𝑥∈𝜆𝐵},𝑥∈span(𝐵). (5) From 𝐸_𝐵 being Banach, we conclude 𝑙₁(𝐵)⊆𝐸_𝐵, but from the proof given in [17, Example 3.2.21], it follows that 𝐵 is not included in span(𝐵).

Let 𝑈 be a connected open subset of a Banach space 𝑋. The space of all the holomorphic functions on 𝑈 is denoted by 𝐻(𝑈). The compact open topology on 𝐻(𝑈) denoted by 𝜏_𝑐. (𝐻(𝑈),𝜏_𝑐) is a semi-Montel space; that is, each closed and bounded subset is compact. A subset 𝐴⊂𝑈 is called 𝑈-bounded if it is bounded and the distance of 𝐴 to the complementary of 𝐴 is positive. For 𝑈=𝑋, 𝑈-bounded means simply bounded. If 𝑈 is the unit ball of a Banach space, 𝐴 is 𝑈-bounded if and only if it is contained in a ball of radius 𝑟<1. If 𝐸 is a Banach space, the space of 𝐸-valued holomorphic functions on 𝑈 is denoted by 𝐻(𝑈,𝐸). We refer to [18] for the precise definitions. A weight 𝑣∶𝑈∶→]0,∞[ is a continuous function which is strictly positive. According to [19], we say that a weight 𝑣 on 𝑈 satisfies the property (I) whenever it is bounded below in each 𝑈-bounded subset 𝐴 of 𝑈. The weighted Banach spaces of holomorphic functions are defined as 𝐻_𝑣(𝑈)∶=𝑓∈𝐻(𝑈)∶sup_𝑥∈𝑈𝑣(𝑥𝑓(𝑥𝐻))<∞_𝑣0(𝑈)(𝑈)}.∶=𝑓∈𝐻∶𝑣𝑓vanishesatinfinityon𝑈-boundedsets (6) Recall that a function 𝑔∶𝑈→ℝ is said to vanish at infinity on 𝑈-bounded sets when for each 𝜀>0 there exists a 𝑈-bounded subset 𝐴 such that |𝑔(𝑥)|<𝜀 for 𝑥∈𝑈⧵𝐴. 𝐻_𝑣(𝑈) is always continuously embedded in (𝐻(𝑈),𝜏_𝑐). If 𝑣 satisfies (𝐼), then 𝐻_𝑣(𝑈) is continuously embedded in the space of holomorphic functions of bounded type 𝐻_𝑏(𝑈), which is 𝐻(𝑈) endowed with the Fréchet topology of uniform convergence on 𝑈-bounded sets.

Analogously, for a Banach space 𝐸, we define the weighted spaces of vector-valued functions as 𝐻_𝑣(𝑈,𝐸)∶=𝑓∈𝐻(𝑈,𝐸)∶sup_𝑥∈𝑈𝑣(𝑥𝑓(𝑥𝐻))<∞_𝑣0()()𝑓.𝑈,𝐸∶=𝑓∈𝐻𝑈,𝐸∶𝑣vanishesatinfinityon𝑈-boundedsets (7) During all the work, our model spaces will be 𝐻_𝑣(𝑈) and 𝐻_𝑣0(𝑈). But we will deal with general closed subspaces 𝐴_𝑣(𝑈) of 𝐻_𝑣(𝑈) and their corresponding vector-valued analogues 𝐴_𝑣(𝑈,𝐸) (which will be defined in the following section) in order to consider important subspaces as they are the spaces 𝒫(^𝑚𝑋) of homogeneous polynomials of degree 𝑚 and, in case of 𝑈 being bounded, the algebras 𝐴(𝑈) and 𝐴_𝑢(𝑈) of holomorphic and bounded functions which are continuous and uniformly continuous on 𝑈, respectively.

Let 𝐴_𝑣(𝑈) be a subspace of 𝐻_𝑣(𝑈). A subset 𝐴⊆𝑈 is said to be a set of uniqueness for 𝐴_𝑣(𝑈) if each 𝑓∈𝐴_𝑣(𝑈) which vanishes at 𝐴 is identically null. A set 𝐴⊆𝑋 is said to be sampling for 𝐴_𝑣(𝑈) if there exists some constant 𝐶≥1 such that, for every 𝑓∈𝐴_𝑣(𝑈), sup_𝑥∈𝑋𝑣(𝑥)𝑓(𝑥)≤𝐶sup_𝑎∈𝐴𝑣(𝑎)𝑓(𝑎). (8)

In case 𝐴_𝑣(𝑈) is an algebra the constant, 𝐶 can be always taken 1 and, according to Globevnik, the sampling sets are called boundaries [20–22]. If 𝐴⊂𝑈 and 𝑀_𝐴={𝑣(𝑥)𝛿_𝑥∶𝑥∈𝐴}⊆𝐴_𝑣(𝑈)^∗, it follows from the definitions that 𝑀_𝐴⊆𝐵_𝐴^∗_𝑣, 𝐴 is sampling if and only if 𝑀_𝐴 is norming, and 𝐴 is a set of uniqueness if and only if 𝑀_𝐴 is total.

The sampling sets (as well as interpolation sets) of the weighted space 𝐴^−𝑝(𝔻) (i.e., 𝐻_𝑣(𝔻) for 𝑣(𝑧)=(1−𝑧)^𝑝, 𝑝>0) were characterized by Seip in [23] in terms of certain densities.

2. Banach Subspaces of 𝐻_𝑣(𝑈) Which Are Dual Spaces

Let one consider 𝐴_𝑣(𝑈)⊆𝐻_𝑣(𝑈) to be a subspace with compact closed unit ball for 𝜏_𝑐. Notice that this condition implies that 𝐴_𝑣(𝑈) is norm closed. We define the Banach space of vector-valued functions in a weak sense: 𝐴_𝑣(𝑈,𝐸)∶=𝑓∶𝑈→𝐸∶𝑢∘𝑓∈𝐴_𝑣(𝑈)∀𝑢∈𝐸^∗. (9) Since weakly holomorphic functions are holomorphic and weakly bounded sets are bounded, it follows that for 𝐴_𝑣(𝑈)=𝐻_𝑣(𝑈) this definition agrees with the strong definition given previously. Following the same steps as in [12, Theorem 2.1] (also in [24, Lemma 10]), we get that 𝐴_𝑣(𝑈,𝐸) can be identified with 𝐿(𝐺_𝐴𝑣,𝐸), 𝐺_𝐴𝑣 being the predual of 𝐴_𝑣(𝑈) that exists by the Dixmier-Ng Theorem [25].

Remark 5. In [25], it is shown that 𝐺_𝐴𝑣 consists of all the functionals 𝑦∈𝐴_𝑣(𝑈)^∗ such that 𝑦 restricted to 𝐵_𝐴𝑣 is 𝜏_𝑐 continuous. Let 𝐴⊆𝑈. If we denote 𝑀_𝐴∶={𝛿_𝑥∶𝑥∈𝐴}⊂𝐵_𝐺𝐴𝑣, we have that 𝑀_𝐴 is separating in 𝐴_𝑣(𝑈) if and only if span(𝑀_𝐴) is (weakly) dense if and only if 𝐴 is a set of uniqueness for 𝐴_𝑣(𝑈). Analogously, 𝑀_𝐴 is norming in 𝐴_𝑣(𝑈) if and only if 𝐴 is sampling for 𝐴_𝑣(𝑈).

Proposition 6. Let 𝐴_𝑣(𝑈) be a subspace of 𝐻_𝑣(𝑈) with 𝜏_𝑐 compact closed unit ball. Then, 𝑓∈𝐴_𝑣(𝑈,𝐸) if and only if there exists 𝑇∈𝐿(𝐺_𝐴𝑣,𝐸) such that 𝑓(𝑥)=𝑇(𝛿_𝑥). Moreover, the correspondence is an isometry.

Proof. If 𝑇∈𝐿(𝐺_𝐴𝑣,𝐸), we define 𝑓(𝑥)∶=𝑇(𝛿_𝑥). Since 𝑇 is continuous and 𝐺^∗_𝑣=𝐴_𝑣(𝑈), it follows that 𝑢∘𝑓∈𝐴_𝑣(𝑈) for each 𝑢∈𝐸^∗, and then, 𝑓∈𝐴_𝑣(𝑈,𝐸) by the very definition.
Conversely, let 𝑓∈𝐴_𝑣(𝑈,𝐸). We set 𝑀_𝑈∶={𝑣(𝑥)𝛿_𝑥∶𝑥∈𝑈}. Since 𝑀_𝑈 is 1-norming in 𝐴_𝑣(𝑈), we apply Lemma 2 and Remark 3, to get that 𝐺_𝐴𝑣=𝑙₁(𝑀)={∑_𝑥∈𝑈𝛼_𝑥𝑣(𝑥)𝛿_𝑥∶∑_𝑥∈𝑈|𝛼_𝑥|<∞}. We see that 𝑇(𝛿_𝑥)∶=𝑓(𝑥) defines a linear mapping on 𝐺_𝑣. If ∑_𝑥∈𝑈𝛼_𝑥𝛿_𝑥=0, then, for each 𝑢∈𝐸^∗, 𝑢,𝑇_𝑥∈𝑈𝛼_𝑥𝛿_𝑥=𝑢,_𝑥∈𝑈𝛼_𝑥𝑓(𝑥=)𝑢∘𝑓,_𝑥∈𝑈𝛼_𝑥𝛿_𝑥=0, (10) and then, 𝑇 is well defined. Moreover, since 𝐵_𝐺𝑣={∑_𝑥∈𝑈𝛼_𝑥𝑣(𝑥)𝛿_𝑥∶∑_𝑥∈𝑈|𝛼_𝑥|<1}, it is easy to compute ‖𝑇‖=sup{𝑣(𝑥)‖𝑓(𝑥)‖∶𝑥∈𝑈}=‖𝑓‖_𝑣.

Now, we are going to show that there are more natural spaces with compact unit ball for the compact open topology. To do this, we present a general result of complemented subspaces in the Fréchet spaces of analytic functions which could be of independent interest. We state it for Fréchet instead of Banach to include the important space 𝐻_𝑏(𝑈). For 𝐻_𝑣(𝑈) and 𝐻_𝑏(𝑈) with 𝑣 radial and 𝑈 balanced, it is done by García et al. in [19, Proposition 3, Example 14].

Theorem 7. Let 𝐹(𝑈) be a Fréchet space of holomorphic functions on 𝑈 such that 𝐹(𝑈)↪(𝐻(𝑈),𝜏_𝑐) continuously. If, for 𝑚∈ℕ, 𝒫(^𝑚𝑋)⊂𝐹(𝑈), then 𝒫(^𝑚𝑋) endowed with its norm topology is a complemented subspace of 𝐹(𝑈). If 𝐹(𝑈)=𝐻_𝑣(𝑈), then 𝐵_𝑣∩𝒫(^𝑚𝑋) is compact in (𝐻(𝑈),𝜏_𝑐).

Proof. Let one assume without loss of generality that 0∈𝑈. For 𝑓∈𝐻(𝑈), we denote 𝑃^𝑓_𝑚∈𝒫(^𝑚𝑋) the 𝑚-homogeneous polynomial such that 𝑓(𝑥)=𝑓(0)+_𝑘∈ℕ𝑃^𝑓_𝑘(𝑥). (11) Let 𝜏_𝑝 be denoted by the topology in 𝒫(^𝑚𝑋) of pointwise convergence on 𝐵_𝑋. The projection 𝑃_𝑚∶𝐻(𝑈),𝜏_𝑐→𝑃^𝑚𝑋,𝜏_𝑝,𝑓↦𝑃^𝑓_𝑚, (12) is continuous. We checked it. Let 𝑓∈𝐻(𝑈), and let (𝑓_𝑖)_𝑖∈𝐼 be a net convergent to 𝑓 in (𝐻(𝑈),𝜏_𝑐). Let 𝑟>0 such that the closed ball 𝐷(0,𝑟)⊂𝐸, and let 𝑢∈𝐸 with ‖𝑢‖=1. For 𝑖∈𝐼, we define 𝑔_𝑖(𝑧)∶=𝑓_𝑖(𝑧𝑢)∈𝐻^∞(𝐵_𝑟), 𝐵_𝑟 being the ball with radius 𝑟 in ℂ. Let 𝑔(𝑧)∶=𝑓(𝑧𝑢). We have that (𝑔_𝑖)_𝑖∈𝐼 converges to 𝑔 in 𝐻^∞(𝐵_𝑟). We conclude from the continuity of the evaluations of the derivatives in this last space and 𝑃^𝑓𝑖_𝑚(𝑢)=𝑔_𝑖^(𝑚)(0)𝑚!𝑚∈ℕ₀,𝑖∈𝐼. (13)
Hence, by the closed graph theorem, we get that the map 𝑃_𝑚∶𝐹(𝑈)→𝒫(^𝑚𝑋), 𝑓↦𝑃^𝑓_𝑚 is continuous. Since the map is by hypothesis surjective and restricted to 𝒫(^𝑚𝑋) that is the identity, it also follows that 𝒫(^𝑚𝑋) is closed in 𝐹(𝑈). Thus, the inclusion 𝑖∶=𝑃_𝑚|_𝒫(^𝑚_𝑋) is an isomorphism. Hence, the inverse of the inclusion 𝑗∶=𝑖⁻¹∶𝒫(^𝑚𝑋)→𝐹(𝑈) satisfies that 𝑃_𝑚∘𝑗 is the identity in 𝒫(^𝑚𝑋). We apply [26, Chapter 2, Section 7, Proposition 3] to conclude that 𝒫(^𝑚𝑋) is complemented in 𝐹(𝑈).
We check now that 𝐵_𝑣∩𝒫(^𝑚𝑋) is compact for the topology of pointwise convergence on 𝑈. Let (𝑓_𝑖)_𝑖∈𝐼 be a net in 𝒫(^𝑚𝑋)∩𝐵_𝑣 such that it is convergent to 𝑓∈𝐵_𝑣 pointwise in 𝑈. Assume without loss of generality that 𝑉∶=𝑈∩𝐵_𝑋 is nonempty. The net (𝑓_𝑖)_𝑖∈𝐼 is a bounded net in 𝒫(^𝑚𝑋) which is Cauchy for the topology of pointwise convergence in 𝑉. This topology is Hausdorff and weaker than the topology of pointwise convergence in 𝐵_𝑋. Since (𝒫(^𝑚𝑋),‖⋅‖_𝑚) is a dual space [18, Proposition 1.17], the topology of pointwise convergence on 𝐵_𝑋 is relatively compact restricted to the bounded sets in (𝒫(^𝑚𝑋),‖⋅‖_𝑚) and then agrees in the bounded sets with the topology of pointwise convergence on 𝑉. Moreover, 𝐵_𝑣∩𝒫(^𝑚𝑋) is bounded in (𝒫(^𝑚𝑋),‖⋅‖_𝑚), and hence, we get that (𝑓_𝑖)_𝑖 is convergent to 𝑔∈𝒫(^𝑚𝑋) pointwise in 𝑋. Since 𝑈⊆𝑋, we get 𝑓=𝑔. We have proved that 𝐵_𝑣∩𝒫(^𝑚𝑋) is closed in 𝐵_𝑣 for the topology of pointwise convergence in 𝑈, and then it is compact.

For spaces 𝐻_𝑣(𝑈) containing 𝒫(^𝑚𝑋), we have that (𝒫(^𝑚𝑋),‖⋅‖_𝑣) is a subspace which is complemented and it is isomorphic to 𝒫(^𝑚𝑋) endowed with its natural norm ‖⋅‖_𝑚. Moreover, (𝒫(^𝑚𝑋),‖⋅‖_𝑣) has a compact unit ball for the topology of pointwise convergence in 𝑈, and hence, it is a dual Banach space because of Dixmier-Ng theorem [25]. We denote by 𝐺^𝑚_𝑣 the predual of (𝒫(^𝑚𝑋),‖⋅‖_𝑣) and by 𝐺^𝑚 the predual of (𝒫(^𝑚𝑋),‖⋅‖_𝑚) obtained in [12, Theorem 2.4]. In 𝐺^𝑚, the subset 𝑀∶={𝛿_𝑥∶𝑥∈𝐵_𝑋} is norming and then spans a (𝜎(𝐺^𝑚,𝒫(^𝑚𝑋))-) dense subspace. The same applies for 𝑀_𝑣∶={𝑣(𝑥)𝛿_𝑥∶𝑥∈𝑈} in (𝒫(^𝑚𝑋),‖⋅‖_𝑣). Both 𝑀 and 𝑀_𝑣 are formed by functionals which are linearly independent by [11, Proposition 1]. We check below that there is a natural isomorphism between 𝐺^𝑚 and 𝐺^𝑚_𝑣.

Proposition 8. Let 𝑣 be a weight on 𝑈 such that 𝒫(^𝑚𝑋)∈𝐻_𝑣(𝑈). The predual 𝐺^𝑚_𝑣 of (𝒫(^𝑚𝑋),‖⋅‖_𝑣) is isomorphic to the predual 𝐺^𝑚 of (𝒫(^𝑚𝑋),‖⋅‖_𝑚) canonically, that is, there exists 𝑇∶𝐺^𝑚_𝑣→𝐺^𝑚 such that 𝑇(𝛿_𝑥)=𝛿_𝑥 for each 𝑥∈𝑈.

Proof. Let 𝑀_𝑣 be as defined previously. If we define𝑇₁∶span(𝑀_v)→𝐺^𝑚, by means of 𝛿_𝑥↦𝛿_𝑥, we have that 𝑇 is well defined since 𝛿_𝑥=‖𝑥‖^𝑚𝛿_{𝑥/‖𝑥‖}∈span(𝑀), it is (weakly) continuous, and then, it can be extended to 𝑇₁∶𝐺^𝑚_𝑣→𝐺^𝑚. If we consider now span(𝑀_𝑣) as a subspace of 𝐺^𝑚 it is (weakly) dense since open sets are sets of uniqueness in 𝒫(^𝑚𝑋). The linear map 𝑇₂∶span(𝑀_𝑣)→𝐺^𝑚_𝑣, 𝛿_𝑥↦𝛿_𝑥 is again (weakly) continuous, and hence that we get, an extension 𝑇₂∶𝐺^𝑚→𝐺^𝑚_𝑣. 𝑇₂∘𝑇₁∶𝐺^𝑚_𝑣→𝐺^𝑚_𝑣 is a continuous linear mapping, and then, it is the identity since both coincide in span(𝑀_𝑣). Moreover, 𝑇₁(𝐺^𝑚_𝑣) has dense range in 𝐺^𝑚. Hence, 𝑇₁ is an onto isomorphism by [26, Chapter 2, Section 7, Proposition 3].

From the linearization of these dual Banach subspaces of 𝐻_𝑣(𝑈), one can get easily an extension of the Blaschke-type result for vector-valued functions [4, Theorem 2.5] generalized in [7, Corollary 4.2]. The proof that we give is strongly based on the Banach-Steinhaus principle.

Proposition 9. Let 𝐴_𝑣(𝑈) be a subspace of 𝐻_𝑣(𝑈) which has a 𝜏_𝑐-compact closed unit ball, let 𝐴⊆𝑈 be a set of uniqueness for 𝐴_𝑣(𝑈), and let 𝐸 be a Banach space. If (𝑓_𝑖)_𝑖∈𝐼 is a bounded net in 𝐴_𝑣(𝑈,𝐸) such that (𝑓_𝑖(𝑥))_𝑖∈𝐼 is convergent for each 𝑥∈𝐴, then (𝑓_𝑖)_𝑖∈𝐼 is convergent to a function 𝑓∈𝐴_𝑣(𝑈,𝐸) uniformly on the compact subsets of 𝑈.

Proof. Let (𝑇_𝑖)_𝑖∈𝐼 be the sequence of operators in 𝐿(𝐺_𝐴𝑣,𝐸) such that 𝑓_𝑖(𝑥)=𝑇_𝑖(𝛿_𝑥) for each 𝑥∈𝑈. Let 𝑀_𝑣={𝑣(𝑥)𝛿_𝑥∶𝑥∈𝑈}⊆𝐺_𝑣(𝑈). 𝑀_𝑣 is a 1-norming subset of 𝐺_𝑣, that is, Γ𝑀_𝑣=𝑀_𝑣^∘∘=𝐵^∘_𝐴𝑣=𝐵_𝐺𝐴𝑣. (14) By hypothesis, there exists 𝐶>0 such that 𝑇_𝑖𝑀_𝑣=𝑣(𝑥)𝑓_𝑖(𝑥)∶𝑥∈𝑈⊂𝐶𝐵_𝐸foreach𝑖∈𝐼. (15) Thus, 𝑇_𝑖𝐵_𝐺𝐴𝑣=𝑇_𝑖Γ𝑀_𝑣⊂Γ𝑇_𝑖𝑀_𝑣⊆𝐶𝐵_𝐸, (16) for each 𝑖∈𝐼. By Remark 5, the subset 𝑀_𝐴∶={𝛿_𝑥∶𝑥∈𝐴} is total in 𝐺_𝐴𝑣. Since (𝑇_𝑖)_𝑖∈𝐼 is equicontinuous, the topology of pointwise convergence on 𝐺_𝐴𝑣 coincides with the topology of pointwise convergence in 𝑀_𝐴 by [27, 39.4(1)]. Thus, (𝑇_𝑖)_𝑖∈𝐼 is pointwise convergent to 𝑇∈𝐿(𝐺_𝐴𝑣,𝐸). The convergence is uniform on the compact subsets of 𝐺_𝐴𝑣 by [27, 39.4(2)]. If 𝐾⊂𝑈 is compact, then {𝛿_𝑥∶𝑥∈𝐾} is compact in 𝐺_𝐴𝑣. This follows from the observation that 𝑈→𝐺_𝐴𝑣, 𝑥↦𝛿_𝑥 is (weakly) holomorphic and then continuous.

Proposition 9 and Theorem 7 yield that the Banach-Steinhaus theorem stated as in [27, 39.4(1)] can be extended to the space of vector-valued polynomials 𝑃(^𝑚𝑋,𝐸). Bochnak and Siciak showed [28, Theorem 2] that the uniform boundedness principle also is valid for polynomials.

The following results are extensions of those obtained in [7] by Frerick et al. for spaces of bounded holomorphic and harmonic functions on open subsets of finite-dimensional subspaces with values in locally convex spaces. Our results are valid for spaces of functions defined on an open and connected subset 𝑈 of a Banach space 𝑋, but we restrict to the case of Banach-valued functions. The proofs that we give here are simpler. The next theorem extends [7, Theorem 2.2].

Theorem 10. Let 𝑣 be a weight on 𝑈, let 𝐴_𝑣(𝑈) be a subspace of 𝐻_𝑣(𝑈) with 𝜏_𝑐-compact closed unit ball, let 𝐴 be a set of uniqueness for 𝐴_𝑣(𝑈), let 𝐸 be a Banach space, and let 𝐻⊆𝐸^∗ be a subspace which determines boundedness in 𝐸. If 𝑓∶𝐴→𝐸 is a function such that 𝑢∘𝑓 admits an extension 𝑓_𝑢∈𝐴_𝑣(𝑈) for each 𝑢∈𝐻, then 𝑓 admits a unique extension 𝐹∈𝐴_𝑣(𝑈,𝐸).

Proof. Let 𝐹_𝐴 be the span of {𝛿_𝑥∶𝑥∈𝐴}. The hypothesis implies that 𝐹_𝐴 is 𝜎(𝐺_𝐴𝑣,𝐴_𝑣(𝑈)) dense, and then, it is dense in norm. The map 𝑇∶𝐹_𝐴→𝐸, 𝛿_𝑥↦𝑓(𝑥) is well defined since 𝐻 is separating. Let 𝑥=∑^𝑘_𝑖=1𝛼_𝑖𝛿_𝑥𝑖 be an element in the unit ball 𝐵_𝐹𝐴 of 𝐹_𝐴, and let 𝑢∈𝐻. We compute: |⟨𝑇(𝑥),𝑢⟩|=^𝑘_𝑖=1𝛼_𝑖𝑓𝑥_𝑖=,𝑢_𝑘𝑖=1𝛼_𝑖𝛿_𝑥𝑖,𝑓_𝑢≤𝑓_𝑢𝑣. (17)
Since this is true for each 𝑥∈𝐵_𝐹𝐴, we conclude that 𝑇(𝐵_𝐹𝐴) is 𝜎(𝐸,𝐻) bounded and then norm bounded by hypothesis. Thus, 𝑇∶𝐹_𝐴→𝐸 is a bounded linear mapping. Since 𝐹_𝐴 is dense in 𝐺_𝐴𝑣, we can extend 𝑇 to 𝑇∶𝐺_𝐴𝑣→𝐸. We conclude by Proposition 6.

The following result is a generalization of [6, Theorem 1(ii)].

Theorem 11. Let 𝑣 be a weight on 𝑈, let 𝐴_𝑣(𝑈) be a subspace of 𝐻_𝑣(𝑈) with 𝜏_𝑐-compact closed unit ball, let 𝐴 be a set of uniqueness for 𝐴_𝑣(𝑈), let 𝐸 be a Banach space, and let 𝐻⊆𝐸^∗ be a norming subspace. If 𝑓∶𝐴→𝐸 is a function such that 𝑢∘𝑓 admits an extension 𝑓_𝑢∈𝐴_𝑣(𝑈) for each 𝑢∈𝐻 such that (𝑓_𝑢)_{𝑢∈𝐵𝐻} is bounded in 𝐴_𝑣(𝑈), then 𝑓 admits a unique extension 𝐹∈𝐴_𝑣(𝑈,𝐸).

Proof. If 𝑢∈𝐵_𝐻 and (𝑢_𝑛)_𝑛⊂𝐵_𝐻 tend to 𝑢, then (𝑓_𝑢𝑛)_𝑛 is a bounded sequence such that (𝑓_𝑢𝑛(𝑥))_𝑛 converges to 𝑢(𝑓(𝑥)) for each 𝑥∈𝐴. Proposition 9 yields that there exists 𝑓_𝑢∈𝐴_𝑣(𝑈) such that (𝑓_𝑢𝑛(𝑥))_𝑛 tends to 𝑓_𝑢(𝑥) for each 𝑥∈𝑈. The conclusion is a consequence of Proposition 1 and Theorem 10.

We now study the problem of extending functions which admit extensions for functionals in a subspace 𝐻 of 𝐸^∗ which we assume only to be 𝜎(𝐸^∗,𝐸) dense. In this case, we require that 𝐴 is quite large. This is symmetric with the problem studied by Gramsch [29], Grosse-Erdmann [8], and Bonet et al. [5]. The next theorem is an extension to our context of [7, Theorem 3.2].

Theorem 12. Let 𝑣 be a weight on 𝑈, let 𝐴_𝑣(𝑈) be a subspace of 𝐻_𝑣(𝑈) with 𝜏_𝑐-compact unit ball, and let 𝐴 be a sampling set for 𝐴_𝑣(𝑈). Let 𝐸 be a Banach space, and let 𝐻 be a 𝜎(𝐸^∗,𝐸)-dense subspace of 𝐸^∗. If 𝑓∶𝐴→𝐸 is a function such sup_𝑎∈𝐴𝑣(𝑎)‖𝑓(𝑎)‖<∞ and such that 𝑢∘𝑓 admits an extension 𝑓_𝑢∈𝐴_𝑣(𝑈) for each 𝑢∈𝐻, then there exists a unique extension 𝐹∈𝐴_𝑣(𝑈,𝐸) of 𝑓.

Proof. The set 𝑀_𝐴∶={𝑣(𝑥)𝛿_𝑥∶𝑥∈𝐴}⊆𝐺_𝐴𝑣 is norming for 𝐴_𝑣(𝑈); hence, we apply Lemma 2 to get that 𝐺_𝐴𝑣 is isomorphic to 𝑙₁(𝑀_𝐴). This means that for each 𝑥∈𝐺_𝐴𝑣, there exists 𝛼∶=(𝛼_𝑛)_𝑛∈𝑙₁ and (𝑥_𝑛)_𝑛⊂𝐴 such that 𝑥=_𝑛𝛼_𝑛𝑣𝑥_𝑛𝛿_𝑥𝑛. (18) The open unit ball 𝐵₁ in 𝐺_𝐴𝑣 for the norm which makes this space isometric to 𝑙₁(𝑀_𝐴) is formed by the vectors 𝑥∈𝐺_𝐴𝑣 such that the sequence (𝛼_𝑛)_𝑛 in the previous representation can be taken in the open unit ball of 𝑙₁. We define that 𝑇∶𝐺_𝐴𝑣→𝐸, 𝑥↦∑_𝑛𝛼_𝑛𝑣(𝑥_𝑛)𝑓(𝑥_𝑛). Since {𝑣(𝑎)𝑓(𝑎)∶𝑎∈𝐴} is bounded by hypothesis, the series is convergent. Moreover, if ∑_𝑛𝛼_𝑛𝑣(𝑥_𝑛)𝛿_𝑥𝑛=0, then for each 𝑢∈𝐻𝑢_𝑛𝛼_𝑛𝑣𝑥_𝑛𝑓𝑥_𝑛=_𝑛𝛼_𝑛𝑣𝑥_𝑛𝛿_𝑥𝑛,𝑓_𝑢=0. (19) Since 𝐻 is separating, 𝑇 is well defined. Moreover, the hypothesis of boundedness of {𝑣(𝑎)𝑓(𝑎)∶𝑎∈𝐴} implies that 𝑇(𝐵₁) is bounded. Hence, we conclude by Proposition 6.

Remark 13. If we consider 𝑓∶𝐵(0,1/2)→𝑙₁, 𝑧→(𝑧^𝑛)_𝑛, we have that 𝑓∈𝐻(𝔻,𝑙₁); hence, 𝑓(𝐵(0,1/2)) is relatively compact in 𝑙₁. Moreover, it is immediate that 𝑢∘𝑓 admits an extension to 𝐻^∞(𝔻) for each 𝑢∈𝜑 (the space of sequences which are zero but finitely many components), 𝜑 is 𝜎(𝑙_∞,𝑙₁) dense (even norming since it is dense in 𝑐₀), and 𝐵(0,1/2) is a set of uniqueness for 𝐻^∞(𝔻). However, 𝑓∉𝐻^∞(𝔻,𝑙₁) since ‖𝑓(𝑧)‖₁=|𝑧|/(1−|𝑧|) for each 𝑧∈𝔻. This shows that the hypothesis in Theorems 10 and 12 is optimal, that is, for the conditions on the set 𝐴 where the functions are defined and in the subspace 𝐻 for which functionals, we have weak extensions that cannot be simultaneously relaxed, and also the condition of boundedness in the extensions in Theorem 11 can not be dropped.

3. General Banach Subspaces of 𝐻_𝑣(𝑈)

For arbitrary Banach spaces 𝐴_𝑣(𝑈)⊂𝐻_𝑣(𝑈) with no assumption on the unit ball, the equivalence between the weak and the strong definitions does not hold in general. We discuss it below. We consider the space 𝐻_𝑣0(𝑈), and we define 𝐻_𝑣0(𝑈,𝐸)_𝑤∶=𝑓∶𝑈→𝐸∶𝑢∘𝑓∈𝐻_𝑣0(𝑈)∀𝑢∈𝐸^∗. (20) A Banach space 𝐸 is said to satisfy the Schur property if every sequence (𝑥_𝑛)_𝑛 in 𝐸 which is weakly convergent is also norm convergent. The well-known theorem of Schur asserts that 𝑙₁ satisfies this property.

Proposition 14. If 𝐸 is a Banach space with the Schur property, then 𝐻_𝑣0(𝑈,𝐸)=𝐻_𝑣0(𝑈,𝐸)_𝑤.

Proof. Suppose that there exists 𝑓∈𝐻_𝑣0(𝑈,𝐸)_𝑤⧵𝐻_𝑣0(𝑈,𝐸). Then, there exist 𝑐>0 and (𝑥_𝑛)_𝑛 going to infinity on 𝑈-bounded sets such that 𝑣(𝑥_𝑛)‖𝑓(𝑥_𝑛)‖>𝑐 and 𝑢∘𝑓∈𝐻_𝑣0(𝑈) for all 𝑢∈𝐸^∗. This last condition implies that 𝑣𝑥_𝑛𝑓𝑥_𝑛∶𝑛∈ℕ (21) is (weakly) convergent to zero, a contradiction.

We see below that the situation differs for function with values in the general Banach spaces.

Example 15. Assume that 𝑋 is finite dimensional and 𝐻_𝑣0(𝑈) is infinite dimensional. Then, 𝐻_𝑣0(𝑈,𝑐₀)⊊𝐻_𝑣0(𝑈,𝑐₀)_𝑤.

Proof. First, we proceed similarly as in [30, Lemma 21] to get a sequence (𝑓_𝑛)_𝑛 in 𝐵_𝑣0 such that (𝑓_𝑛)_𝑛 converges to 0 in 𝜏_𝑐, and there exists 𝛿>0 such that ‖𝑓_𝑛‖_𝑣≥𝛿 for all 𝑛∈ℕ. Since 𝐻_𝑣0(𝑈) is infinite dimensional, there is 𝛿>0 and (𝑔_𝑛)_𝑛∈𝐵_𝑣0 such that ‖𝑔_𝑛−𝑔_𝑘‖_𝑣>𝛿 for 𝑛≠𝑘. We apply that (𝐻_𝑣(𝑈),𝜏_𝑐) is metrizable and 𝐵_𝑣  is 𝜏_𝑐 compact to get that (𝐵_𝑣0,𝜏_𝑐) is relatively sequentially compact. Hence, we can extract a subsequence of (𝑔_𝑛)_𝑛 which is Cauchy for 𝜏_𝑐, and we denote again by (𝑔_𝑛)_𝑛. Defining 𝑓_𝑛∶=(𝑔_𝑛−𝑔_𝑛+1)/2, we get the desired sequence.
We consider 𝑓∶𝑈→𝑐₀, 𝑧→(𝑓_𝑛(𝑧))_𝑛. Let 𝑢=(𝑢_𝑛)_𝑛∈𝑙₁ be arbitrary. Since (𝑓_𝑛)_𝑛⊂𝐵_𝑣0, the series ∑_𝑛𝑢_𝑛𝑓_𝑛 is convergent in 𝐻_𝑣0(𝑈). Hence, 𝑓∈𝐻_𝑣0(𝑈,𝑐₀)_𝑤. The convergence of (𝑓_𝑛)_𝑛 for the compact open topology implies that for each 𝐾⊂𝑈 there exists 𝑛₀=𝑛₀(𝐾) such that sup𝑓_𝑛0(𝑧<𝛿)∶𝑧∈𝐾{𝑣(𝑥)}.2max∶𝑥∈𝐾 (22) Since ‖𝑓_𝑛0‖_𝑣≥𝛿, we obtain that there exists 𝑧₀∈𝑈⧵𝐾 such that 𝑣𝑧₀𝑓𝑧₀≥𝑣𝑧₀𝑓_𝑛0𝑧₀≥𝛿2. (23) Thus, 𝑓∉𝐻_𝑣0(𝑈,𝑐₀).

Example 16. Assume that 𝑈 is the unit ball of a Banach space 𝑋, 𝑔∶[0,1]→]0,∞[ continuous with 𝑔(𝑡)>0 for 0≤𝑡<1 and 𝑔(1)=0 and 𝑣(𝑥)=𝑔(‖𝑥‖) for 𝑥∈𝑈. Then, 𝐻_𝑣0(𝑈,𝑐₀)_𝑤⊊𝐻_𝑣0(𝑈,𝑐₀).

Proof. The hypothesis on 𝑣 implies that for each 𝑓∈𝐻_𝑣 the Taylor polynomials 𝑃^𝑓_𝑛 of the development at zero converge to 𝑓 in (𝐻(𝑈),𝜏_𝑏). If we consider the Cesàro means 𝐶_𝑛𝑓=1𝑛+1^𝑛_𝑖=0^𝑖_𝑘=0𝑃_𝑛𝑓, (24) then 𝐶_𝑛(𝑓)∈𝐵_𝑣0 for each 𝑓∈𝐵_𝑣 ([31, Proposition 1.2], [19, Proposition 4]) and 𝐶_𝑛(𝑓)→𝑓 in (𝐻(𝑈),𝜏_𝑏). If 𝑓∈𝐵_𝑣⧵𝐵_𝑣0, then (𝐶_𝑛(𝑓))_𝑛 is not Cauchy in 𝐻_𝑣(𝑈), since 𝐻_𝑣0(𝑈) is closed. Hence, there are 𝜀>0 and a subsequence (𝑔_𝑘)_𝑘∶=(𝐶_𝑛𝑘(𝑓))_𝑘 such that 𝑔_𝑘−𝑔_𝑘+1𝑣>𝜀. (25) Defining ℎ_𝑘∶=𝑔_𝑘−𝑔_𝑘+1, we have that (ℎ_𝑘)_𝑘 tends to 0 in 𝐻_𝑏(𝑈). Proceeding as in Example 15, we obtain that ℎ∶𝑈→𝑐₀,𝑥↦(ℎ_𝑘(𝑥))_𝑘 satisfies ℎ∈𝐻_𝑣0(𝑈,𝑐₀)_𝑤⧵𝐻_𝑣0(𝑈,𝑐₀).
The proof is complete since 𝐻_𝑣(𝑈)⧵𝐻_𝑣0(𝑈) is never empty. We checked it. If 𝑈=𝔻, then 𝐻_𝑣(𝔻) is the bidual of 𝐻_𝑣0(𝔻) and this last space is not reflexive [32, 33]. Hence, there exists 𝑓₀∈𝐻_𝑣(𝔻)⧵𝐻_𝑣0(𝔻) with ‖𝑓₀‖_𝑣≤1. For 𝑈 arbitrary, we consider 𝑥₀∈𝑋 such that ‖𝑥₀‖=1 and 𝑢∈𝑋^∗ such that ‖𝑢‖=1 and 𝑢(𝑥₀)=1. Define 𝑓∶𝑈→ℂ by 𝑓(𝑥)=𝑓₀(𝑢(𝑥)). Since 𝑔 is nonincreasing, we have 𝑣(𝑥)𝑓(𝑥)=𝑔(‖𝑥‖)𝑓₀(𝑢(𝑥𝑢(𝑥𝑓))≤𝑔(|)|)₀(𝑢(𝑥))≤1. (26) Since 𝑓₀∉𝐻_𝑣0(𝔻), there exist 𝑐>0 and a sequence (𝑧_𝑛)_𝑛 of complex numbers smaller than 1 such that lim_𝑛|𝑧_𝑛|=1 and 𝑣𝑧_𝑛𝑥₀𝑓𝑧_𝑛𝑥₀=𝑔𝑧_𝑛𝑓₀𝑧_𝑛>𝑐, (27) hence, 𝑓∉𝐻_𝑣0(𝑈).

Thus, on the contrary that with the concrete examples of dual spaces 𝐴_𝑣(𝑈) considered in the previous section (𝐻_𝑣(𝑈) and 𝒫(^𝑚𝑋)), in 𝐻_𝑣0(𝑈) the definition of the corresponding spaces of vector-valued functions in the weak sense are not consistent with the natural definition. For linearization for these spaces with the weak definition, we refer to the work of Carando and Zalduendo [11].

In view of Proposition 14, one could expect that the analogous extensions of Theorems 10 and 12 are possible for 𝐻_𝑣0(𝑈,𝐸) when 𝐸 is required to have the Schur property. This is not the case as the following example shows.

Example 17. Let 𝑈 be the unit ball of a Banach space 𝑋, and let 𝑣(𝑥)=1−‖𝑥‖ for 𝑥∈𝑈. Fix 𝑥₀∈𝑋 with ‖𝑥₀‖=1 and ℎ∈𝐵_𝑋^∗ with ℎ(𝑥₀)=1. Consider that 𝑓∶𝑈∶→𝑙₁, 𝑥↦(ℎ(𝑥)^𝑛)_𝑛, then the following applies. (a)𝑓∈𝐻_𝑣(𝑈,𝑙₁) and for each 0<𝑟<1𝑓_𝑣=sup_{𝑥∈𝑈⧵𝑟𝑈}𝑣(𝑥)𝑓(𝑥)₁=1. (28) Hence, 𝑓∉𝐻_𝑣0(𝑈,𝑙₁). (b)𝑢∘𝑓∈𝐻_𝑣0(𝑈) for each 𝑢=(𝑢_𝑛)_𝑛∈𝑐₀.

Proof. To prove (a), we observe that 𝑝(𝑡)=𝑡/(1−𝑡) is increasing for 𝑡∈[0,1[; hence, sup_𝑥∈𝑈𝑣(𝑥)𝑓₁=sup_𝑥∈𝑈(‖𝑥|ℎ(𝑥1−‖))|1−|ℎ(𝑥)|=sup_{𝑥=𝑡𝑥0,0<𝑡<1}(‖𝑥|ℎ(𝑥1−‖))|1−|ℎ(𝑥)|=sup_{𝑥=𝑡𝑥0,0<𝑡<1}|ℎ(𝑥)|=sup_{𝑥=𝑡𝑥0,𝑟<𝑡<1}(‖𝑥|ℎ(𝑥1−‖))|1−|ℎ(𝑥)|=1. (29) Let 𝑢=(𝑢_𝑛)_𝑛∈𝑐₀. Let 𝜀>0 and 𝑛₀∈ℕ such that 𝑛>𝑛₀ implies |𝑢_𝑛|<𝜀/2. For each 𝑥∈𝑈, since |ℎ(𝑥)|<1, we have 𝑣(𝑥)_𝑛>𝑛0𝑢_𝑛ℎ(𝑥)^𝑛≤(‖𝑥𝜀1−‖)2_𝑛∈ℕ|ℎ(𝑥)|^𝑛≤𝜀2. (30) Let 0<𝑟<1 such that for each 𝑟<𝑡<1(1−𝑡)_𝑛≤𝑛0𝑢_𝑛<𝜀2. (31) From (30), (31), and |ℎ(𝑥)|<1, we obtain that sup_{𝑥∈𝑈⧵𝑟𝑈}𝑣(𝑥)𝑢∘𝑓(𝑥)=sup_{𝑥∈𝑈⧵𝑟𝑈}(1−‖𝑥‖)_𝑛∈ℕ𝑢_𝑛ℎ(𝑥)^𝑛≤𝜀. (32)

Remark 18. The same computation as in Example 17(b) shows that for 1<𝑝<∞ and 𝑣(𝑥)=(1−‖𝑥‖)^(1/𝑝) the function 𝑓∶𝑈→𝑙_𝑝, 𝑥↦(ℎ(𝑥)^𝑛)_𝑛 satisfies that 𝑓∈𝐻_𝑣0(𝑈,𝑙_𝑝)_𝑤⧵𝐻_𝑣0(𝑈,𝑙_𝑝).

4. Spaces of Weighted Compact Range Vector-Valued Holomorphic Functions

In this section, we consider the natural extension to the weighted case of the vector-valued compact holomorphic functions introduced by Aron and Schottenloher in [34] by means of the weak definition, that is, for an open and connected subset 𝑈 of a Banach space 𝑋, a closed subspace 𝐴_𝑣(𝑈) of 𝐻_𝑣(𝑈), and a Banach space 𝐸, we define that 𝐴^𝑐_𝑣(𝑈,𝐸)=𝑓∈𝐻_𝑣(𝑈,𝐸)∶𝑣𝑓(𝑈)isrelativelycompactand𝑢∘𝑓∈𝐴_𝑣(𝑈)∀𝑢∈𝐸^∗. (33)

In case 𝑋 is finite dimensional, the space 𝐻_𝑣0(𝑈,𝐸) is the space of holomorphic functions such that 𝑓 is continuous in the Alexandroff compactification 𝑈∪{∞} of 𝑈 and 𝑓(∞)=0. Hence, 𝐻_𝑣0(𝑈,𝐸)=𝐻^𝑐_𝑣0(𝑈,𝐸) in this case. If 𝑋 is infinite dimensional, the inclusion 𝐻^𝑐_𝑣0(𝑈,𝐸)⊂𝐻_𝑣0(𝑈,𝐸) is strict in general. Observe that if 𝑈 is the unit ball and 𝑣 vanishes at ∞ on 𝑈, then 𝐼|_𝑈∈𝐻_𝑣0(𝑈,𝑋).

We check that this (weak) definition agrees with the natural definition when 𝑈 is the unit ball of 𝑋, 𝑣=1, and 𝐴_𝑣(𝑈)=𝐴_𝑢(𝑈) the space of the holomorphic and uniformly continuous functions on 𝑈, that is, we want to show that 𝐴^𝑐_𝑢()()(𝑈)=𝑈,𝐸∶=𝑓∈𝐻𝑈,𝐸∶𝑓isuniformlycontinuousand𝑓isrelativelycompact𝑓∶𝑈→𝐸∶𝑢∘𝑓∈𝐴_𝑢(𝑈)foreach𝑢∈𝐸^∗𝑓(𝑈).andisrelativelycompact (34)

Assume that 𝑓∈𝐻(𝑈,𝐸) satisfies that 𝑢∘𝑓∈𝐴_𝑢(𝑈) for each 𝑢∈𝐸^∗. Given 𝜀>0, since 𝑓(𝑈) is relatively compact, there exists a weak neighbourhood 𝑉 of 0 such that 𝑓(𝑈)−𝑓(𝑈)∩𝑉⊆𝑓(𝑈)−𝑓(𝑈)∩𝐵(0,𝜀). (35) Let one assume that 𝑉∶={𝑒∈𝐸∶|𝑢_𝑖(𝑒)|<𝛾,1≤𝑖≤𝑘}. Since 𝑢_𝑖∘𝑓 is uniformly continuous on 𝑈 for 1≤𝑖≤𝑘, there exists 𝛿>0 such that ‖𝑥−𝑦‖<𝛿 implies 𝑓(𝑥)−𝑓(𝑦)∈𝑉, and therefore, ‖𝑓(𝑥)−𝑓(𝑦)‖<𝜀.

Given two locally convex spaces 𝐹 and 𝐸, we denote by 𝐹𝜀𝐸 its 𝜀-product of Schwartz, that is, the space of all linear and continuous mappings ℒ_𝑒(𝐹^′_co,𝐸), endowed with the topology of uniform convergence on the equicontinuous subsets of 𝐹^′.  𝐹^′_co is 𝐹^′ endowed with the topology 𝜏_co of uniform convergence on the convex compact subsets of 𝐹. The 𝜀-product is symmetric by means of the transpose mapping [27, 43.3(3)]. In case 𝐸 and 𝐹 being Banach spaces, 𝑇∶𝐹^∗→𝐸 belongs to 𝐹𝜀𝐸 if and only if 𝑇 is a compact operator which is weak*-weak continuous by [27, 43.3(2)]. The next theorem is the analogous of Theorem 12 in the case of general Banach spaces of functions, not necessarily dual Banach spaces. However, the techniques used here are different. The proof is analogous to the one given by Bierstedt and Holtmanns in [35] when the linearization result is obtained in a much more general context, but we only require the function to be defined in a sampling set.

Theorem 19. Let 𝐴_𝑣(𝑈) be a closed subspace of 𝐻_𝑣(𝑈), and let 𝐴⊂𝑈 be a sampling set for 𝐻_𝑣(𝑈). Let 𝐸 be a Banach space, and let 𝐻 be a weak*-dense subspace of 𝐸^∗. The following are equivalent. (i)𝑓∶𝐴→𝐸 satisfies that (𝑣𝑓)(𝐴) is relatively compact in 𝐸, and 𝑢∘𝑓 admits an extension 𝑓_𝑢∈𝐴_𝑣(𝑈) for each 𝑢∈𝐻. (ii)The linear mapping 𝑇∶𝐻→𝐴_𝑣(𝑈),  𝑢→𝑓_𝑢 admits an extension 𝑇∈𝐴_𝑣(𝑈)𝜀𝐸. (iii)𝑓 can be extended to 𝐹∈𝐴^𝑐_𝑣(𝑈,𝐸).

Proof. If 𝑓 satisfies (i), then the linear mapping 𝑇∶𝐻→𝐴_𝑣(𝑈),  𝑢↦𝑓_𝑢 is 𝜏_co - ‖⋅‖ continuous, since the absolute convex hull of 𝑓(𝐴) is relatively compact in 𝐸, and the uniform convergence on 𝐴 defines an equivalent topology in 𝐴_𝑣(𝑈). By the Hahn-Banach theorem, 𝐻 is dense in 𝐸^∗ endowed with any topology 𝑡 such that 𝐸=(𝐸^∗,𝜎(𝐸^∗,𝐸))^′=(𝐸^∗,𝑡)^′, that is, which respects the duality of 𝐸^∗ and 𝐸, in particular, for 𝐸^∗_co∶=(𝐸^∗,𝜏_co). Since 𝐴_𝑣(𝑈) is a Banach space, we can extend it to 𝑇∈ℒ(𝐸^∗_co,𝐴_𝑣(𝑈))=𝐸𝜀𝐴_𝑣(𝑈).
If (ii) is satisfied, then the transpose 𝑇^𝑡∶𝐴_𝑣(𝑈)^∗→𝐸 is weak*-weak continuous, and 𝑇^𝑡 maps the unit ball of 𝐴_𝑣(𝑈)^∗ to a relatively compact subset of 𝐸 ([27, 43.3(2),(3)]). We define 𝑓(𝑥)=𝑇^𝑡(𝛿_𝑥), since 𝑢∘𝑓(𝑥)=⟨𝑇(𝑢),𝛿_𝑥)⟩ for each 𝑥∈𝑈, and for each 𝑢∈𝐸^∗, we have that 𝑢∘𝑓∈𝐴_𝑣(𝑈). We conclude since {𝑣(𝑥)𝛿_𝑥∶𝑥∈𝑈} is in the unit ball of 𝐴_𝑣(𝑈)^∗.
Finally, that (iii) implies (i) is trivial.

Observe that, setting 𝐴=𝑈 in Theorem 19, we obtain a linearization of the space 𝐴^𝑐_𝑣(𝑈,𝐸), which also can be obtained as a consequence of the much more general linearization result given by Bierstedt in [14, Bemerkung 3.1] and [15, Corollary 3.94]. Example 15 shows that Theorems 10 and 12 cannot be stated avoiding the condition of relative compactness on the range for general Banach spaces of holomorphic functions.

We finish showing that the weak definition given in this section for 𝐻^𝑐_𝑣0(𝑈,𝐸) is consistent with the natural one, that is, 𝐻^𝑐_𝑣0(𝑈,𝐸)=𝑓∈𝐻_𝑣0()∶(𝑈)=𝑈,𝐸𝑣𝑓isrelativelycompact𝑓∈𝐻_𝑣()∶(𝑈)𝑈,𝐸𝑣𝑓isrelativelycompactand𝑢∘𝑓∈𝐻_𝑣0(𝑈)∀𝑢∈𝐸^∗. (36)

We use a similar argument to the one used by Bierstedt in [13, page 200] in a more general setting, including our case when 𝑋 is finite dimensional (i.e., putting compact instead of 𝑈-bounded in the definition of 𝐻_𝑣0(𝑈)). If 𝑓∶𝑈→𝐸 satisfies that 𝑢∘𝑓∈𝐻_𝑣0(𝑈) for each 𝑢∈𝐸^∗, then by the previous theorem, there exists 𝑇∶𝐻_𝑣0(𝑈)^∗→𝐸 defined by 𝑇(𝛿_𝑥)=𝑓(𝑥) which is weak*-weak continuous and such that 𝑇(𝐵_𝑣0) is relatively compact. This implies that the restriction of 𝑇 to 𝐵_𝑣0 is weak*-norm continuous. Let 𝜀>0. There exists a weak* 0-neighbourhood 𝑉 in 𝐻_𝑣0(𝑈)^∗ such that ‖𝑇(𝑦)‖<𝜀 for every 𝑦∈𝑉. Let {𝑓₁,…,𝑓_𝑘}⊆𝐻_𝑣0(𝑈) be such that 𝑉={𝑢∈𝐻_𝑣0(𝑈)^∗∶|𝑢(𝑓_𝑖)|<1,1≤𝑖≤𝑘}. There exists a 𝑈-bounded subset 𝐾 such that 𝑣(𝑥)|𝑓_𝑖(𝑥)|≤1 for each 𝑥∈𝑈⧵𝐾. This yields that {𝑣(𝑥)𝛿_𝑥∶𝑥∈𝑈⧵𝐾}∈𝑉, and consequently, 𝑣(𝑥)‖𝑓(𝑥)‖=‖𝑇(𝑣(𝑥)𝛿_𝑥)‖≤𝜀 for every 𝑥∈𝑈⧵𝐾.

Acknowledgments

The author wants to thank J. Bonet for several references, discussions, and ideas provided, which were very helpful and in particular allowed him to prove Theorem 7, Proposition 8, and Examples 15 and 16. Remark 4 is due to him. The participation of M. J. Beltrán in a lot of discussions during all the work has also been very important. Her ideas are also reflected in the paper. The author is also indebted to L. Frerick and J. Wengenroth for communicating to him Lemma 2. The remarks and corrections of the referee have been also really helpful to the final version. The author thanks him/her for that.  This research was partially supported by MEC and FEDER Project MTM2010-15200, GV Project ACOMP/2012/090, and Programa de Apoyo a la Investigacin y Desarrollo de la UPV PAID-06-12.