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Demilinear duality
Journal of Inequalities and Applications volume 2011, Article number: 128 (2011)
Abstract
As is well known, there exist nonlocally convex spaces with trivial dual and therefore the usual duality theory is invalid for this kind of spaces. In this article, for a topological vector space X, we study the family of continuous demilinear functionals on X, which is called the demilinear dual space of X. To be more precise, the spaces with nontrivial demilinear dual (for which the usual dual may be trivial) are discussed and then many results on the usual duality theory are extended for the demilinear duality. Especially, a version of AlaogluBourbaki theorem for the demilinear dual is established.
1 Introduction
Let \mathbb{K}\in \left\{\mathbb{R},\u2102\right\} and X be a locally convex space over \mathbb{K} with the dual X'. There is a beautiful duality theory for the pair (X, X') (see [[1], Chapter 8]). However, it is possible that X' = {0} even for some Fréchet spaces such as L^{p} (0, 1) for 0 < p < 1. Then the usual duality theory would be useless and hence every reasonable extension of X' will be interesting.
Recently, {\mathcal{L}}_{\gamma ,U}\left(X,Y\right), the family of demilinear mappings between topological vector spaces X and Y is firstly introduced in [2]. {\mathcal{L}}_{\gamma ,U}\left(X,Y\right) is a meaningful extension of the family of linear operators. The authors have established the equicontinuity theorem, the uniform boundedness principle and the BanachSteinhaus closure theorem for the extension {\mathcal{L}}_{\gamma ,U}\left(X,Y\right). Especially, for demilinear functionals on the spaces of test functions, Ronglu Li et al have established a theory which is a natural generalization of the usual theory of distributions in their unpublished paper "Li, R, Chung, J, Kim, D: Demidistributions, submitted".
Let X,Y be topological vector spaces over the scalar field \mathbb{K} and \mathcal{N}\left(X\right) the family of neighborhoods of 0 ∈ X. Let
Definition 1.1 [2, Definition 2.1] A mapping f: X → Y is said to be demilinear if f(0) = 0 and there exists γ ∈ C(0) and U\in \mathcal{N}\left(X\right) such that every x ∈ X, u ∈ U and t\in \left\{t\in \mathbb{K}:\mid t\mid \le 1\right\} yield r,s\in \mathbb{K} for which r  1 ≤  γ (t) , s ≤  γ (t) and f(x + tu) = rf(x) + sf(u).
We denote by {\mathcal{L}}_{\gamma ,U}\left(X,Y\right) the family of demilinear mappings related to γ ∈ C(0) and U\in \mathcal{N}\left(X\right), and by {\ud4a6}_{\gamma ,U}\left(X,Y\right) the subfamily of {\mathcal{L}}_{\gamma ,U}\left(X,Y\right) satisfying the following property: if x ∈ X, u ∈ U and t ≤ 1, then f(x + tu) = rf(x) + sf(u) for some s with s ≤  γ (t). Let
which is called the demilinear dual space of X. Obviously, X' ⊂ X^{(γ, U)}.
In this article, first we discuss the spaces with nontrivial demilinear dual, of which the usual dual may be trivial. Second we obtain a list of conclusions on the demilinear dual pair (X, X^{(γ, U)}). Especially, the AlaogluBourbaki theorem for the pair (X, X^{(γ, U)}) is established. We will see that many results in the usual duality theory of (X, X') can be extended to (X, X^{(γ, U)}).
Before we start, some existing conclusions about {\mathcal{L}}_{\gamma ,U}\left(X,Y\right) are given as follows. In general, {\mathcal{L}}_{\gamma ,U}\left(X,Y\right) is a large extension of L(X, Y). For instance, if ·: X → [0, +∞) is a norm, then \parallel \cdot \parallel \phantom{\rule{2.77695pt}{0ex}}\in {\mathcal{L}}_{\gamma ,X}\left(X,\mathbb{R}\right) for every γ ∈ C(0). Moreover, we have the following
Proposition 1.2 ([2, Theorem 2.1]) Let X be a nontrivial normed space, C > 1, δ > 0 and U ={u ∈ X : u ≤ δ}, γ(t) = Ct for t\in \mathbb{K}. If Y is nontrivial, i.e.,Y≠{0}, then the family of nonlinear mappings in {\mathcal{L}}_{\gamma ,U}\left(X,Y\right) is uncountable, and every nonzero linear operator T: X → Y produces uncountably many of nonlinear mappings in {\mathcal{L}}_{\gamma ,U}\left(X,Y\right).
Definition 1.3 A family Г ⊂ Y^{X} is said to be equicontinuous at x ∈ X if for every W\in \mathcal{N}\left(Y\right), there exists V\in \mathcal{N}\left(X\right) such that f(x + V) ⊂ f(x) + W for all f ∈ Г, and Г is equicontinuous on X or, simply, equicontinuous if Г is equicontinuous at each x ∈ X.
As usual, Г ⊂ Y^{X} is said to be pointwise bounded on X if {f(x): f ∈ Г} is bounded at each x ∈ X, and f : X → Y is said to be bounded if f(B) is bounded for every bounded B ⊂ X.
The following results are substantial improvements of the equicontinuity theorem and the uniform boundedness principle in linear analysis.
Theorem 1.4 ([2, Theorem 3.1]) If X is of second category and \mathrm{\Gamma}\subset {\mathcal{L}}_{\gamma ,U}\left(X,Y\right) is a pointwise bounded family of continuous demilinear mappings, then Г is equicontinuous on X.
Theorem 1.5 ([2, Theorem 3.3]) If x is of second category and \mathrm{\Gamma}\subset {\mathcal{L}}_{\gamma ,U}\left(X,Y\right) is a pointwise bounded family of continuous demilinear mappings, then Г is uniformly bounded on each bounded subset of X, i.e.,{f(x): f ∈ Г, x ∈ B} is bounded for each bounded B ⊂ X.
If, in addition, X is metrizable, then the continuity of f ∈ Г can be replaced by boundedness of f ∈ Г.
2 Spaces with nontrivial demilinear dual
Lemma 2.1 Let f\in {\mathcal{L}}_{\gamma ,U}\left(X,\mathbb{K}\right). For each x∈ X, u ∈ U and t ≤ 1, we have
Proof. Since f\in {\mathcal{L}}_{\gamma ,U}\left(X,\mathbb{K}\right), for each x ∈ X, u ∈ U and t ≤ 1, we have f(x + tu) = rf(x) + sf(u) where r  1 ≤ γ(t) and s ≤ γ(t). Then
which implies (2). Then (1) holds by letting x = 0 in (2).
Theorem 2.2 Let X be a topological vector space and f : X → [0, +∞) a function satisfying
Then, for every γ ∈ C(0) andU\in \mathcal{N}\left(X\right), the following (I), (II), and (III) are equivalent:
(I) f\in {\mathcal{L}}_{\gamma ,U}\left(X,\mathbb{R}\right) ;
(II) f(tu) ≤ γ(t)f(u) whenever u ∈ U and t ≤ 1;
(III)f\in {\ud4a6}_{\gamma ,U}\left(X,\mathbb{R}\right).
Proof. (I) ⇒ (II). By Lemma 2.1.

(II)
⇒ (III). Let x ∈ X, u ∈ U and t ≤ 1. Then
f\left(x\right)\mid \gamma \left(t\right)\mid f\left(u\right)\le f\left(x\right)f\left(tu\right)\le f\left(x+tu\right)\le f\left(x\right)+f\left(tu\right)\le f\left(x\right)+\mid \gamma \left(t\right)\mid f\left(u\right).
Define φ : [γ(t), γ(t)] → ℝ by φ(α) = f(x) + αf(u). Then φ is continuous and
So there is s ∈[γ(t), γ(t)] such that f(x + tu) = γ(s) = f(x) + sf(u).

(III)
⇒ (I). {\ud4a6}_{\gamma ,U}\left(X,\mathbb{R}\right)\subset {\mathcal{L}}_{\gamma ,U}\left(X,\mathbb{R}\right).
In the following Theorem 2.2, we want to know whether a paranorm on a topological vector space X is in {\ud4a6}_{\gamma ,U}\left(X,\mathbb{R}\right) for some γ and U. However, the following example shows that this is invalid.
Example 2.3 Let ω be the space of all sequences with the paranorm·:
Then, for every γ ∈ C(0) and U_{ ε } = {u = (u_{ j } ): u < ε}, we have\parallel \cdot \parallel \phantom{\rule{2.77695pt}{0ex}}\notin {\mathcal{L}}_{\gamma ,U}\left(\omega ,\mathbb{R}\right). Otherwise, there exists γ ∈ C(0) and ε > 0 such that\parallel \cdot \parallel \phantom{\rule{2.77695pt}{0ex}}\notin {\mathcal{L}}_{\gamma ,U}\left(\omega ,\mathbb{R}\right)and hence
by Theorem 2.2. Pick N ∈ ℕ with\frac{1}{{2}^{N}}<\epsilon. Let{u}_{n}=\left(0,\cdots \phantom{\rule{0.3em}{0ex}},0,\stackrel{\left(N\right)}{n},0,\cdots \phantom{\rule{0.3em}{0ex}}\right), ∀n ∈ ℕ. Then\parallel {u}_{n}\parallel =\frac{1}{{2}^{N}}\frac{n}{1+n}<\frac{1}{{2}^{N}}<\epsilonimplies u_{ n } ∈ U_{ ε } for each N ∈ ℕ. It follows from
that\gamma \left(\frac{1}{n}\right)\nrightarrow 0as n → ∞, which contradicts γ ∈ C(0).
Note that the space ω in Example 2.3 has a Schauder basis. The following corollary shows that the set of nonlinear demilinear continuous functionals on a Hausdorff topological vector space with a Schauder basis has an uncountable cardinality.
Corollary 2.4 Let X be a Hausdorff topological vector space with a Schauder basis. Then for every γ ∈ C(0) and U\in \mathcal{N}\left(X\right), the demilinear dual {X}^{\left(\gamma ,U\right)}=\left\{f\in {\mathcal{L}}_{\gamma ,U}\left(X,\mathbb{R}\right):f\phantom{\rule{2.77695pt}{0ex}}is\phantom{\rule{2.77695pt}{0ex}}continuous\right\} is uncountable.
Proof. Let {b_{ k } } be a Schauder basis of X. There is a family P of nonzero paranorms on X such that the vector topology on X is just σP, i.e., x_{ α } → x in X if and only if x_{ α }  x → 0 for each · ∈ P ([[1], p.55]).
Pick · ∈ P. Then \parallel {\sum}_{k=1}^{\infty}{s}_{k}{b}_{k}\parallel \ne 0 for some {\sum}_{k=1}^{\infty}{s}_{k}{b}_{k}\in X and hence \parallel {s}_{{k}_{0}}{b}_{{k}_{0}}\parallel \ne 0 for some k_{0} ∈ ℕ. For nonzero c\in \mathbb{K}, define f_{ c } : X → [0, +∞) by
Obviously, f_{ c } is continuous and satisfies the condition (*) in Theorem 2.2. Let γ ∈ C(0), {\sum}_{k=1}^{\infty}{r}_{k}{b}_{k}\in X and t ≤ 1. Then
and hence {f}_{c}\in {\ud4a6}_{\gamma ,U}\left(X,\mathbb{R}\right)\subset {\mathcal{L}}_{\gamma ,U}\left(X,\mathbb{R}\right) for all U\in \mathcal{N}\left(X\right) by Theorem 2.2. Thus, \left\{{f}_{c}:0\ne c\in \mathbb{K}\right\}\subset {X}^{\left(\gamma ,U\right)} for all γ ∈ C(0) and U\in \mathcal{N}\left(X\right).
Example 2.5 As in Example 2.3, the space (ω, ·) is a Hausdorff topological vector space with the Schauder base \left\{{e}_{n}=\left(0,\cdots \phantom{\rule{0.3em}{0ex}},0,\stackrel{\left(n\right)}{1},0,\cdots \phantom{\rule{0.3em}{0ex}}\right):n\in \mathbb{N}\right\} . Define f_{ c,n } : ω → ℝ with f_{ c,n } (u) = cu_{ n }  where u = (u_{ j } ) ∈ ω. Then we have
for every γ ∈ C(0) and U\in \mathcal{N}\left(\omega \right)by Corollary 2.4.
Recall that a pseminorm · (0 < p ≤ 1) on a vector space E is characterized by x ≥ 0, tx = t ^{p} x and x + y ≤ x + y for all t\in \mathbb{K} and x, y ∈ E. If, in addition, x = 0 implies x = 0, then, · is called a pnorm on E.
Definition 2.6 ([[3], p. 11][[4], Sec. 2]) A topological vector space X is semiconvex if and only if there is a family {p_{ α } } of (continuous) k_{ α }seminorms ( 0 < k_{ α } ≤ 1) such that the sets {x ∈ X : p_{ α } (x) < 1} form a neighborhood basis at 0, that is,
is a base of\mathcal{N}\left(X\right), where P is the family of all continuous pseminorms with 0 < p ≤ 1.
A topological vector space X is locally bounded if and only if its topology is given by a pnorm (0 < p ≤ 1) ([[5], §15, Sec. 10]).
Clearly, locally bounded spaces and locally convex spaces are both semiconvex.
Corollary 2.7 Let X be a semiconvex Hausdorff topological vector space and p_{0}a continuous k_{0}seminorm ( 0 < k_{0} ≤ 1) on X. Then for {U}_{0}=\left\{x\in X:{p}_{0}\left(x\right)\le 1\right\}\in \mathcal{N}\left(X\right) and \gamma \left(\cdot \right)=e\mid \cdot {\mid}^{{\mathbb{K}}_{0}}\in {\mathbb{K}}^{\mathbb{K}}, the demilinear dual
is uncountable. Especially, \left\{{p}_{0}\left(\cdot \right),\phantom{\rule{2.77695pt}{0ex}}\mathsf{\text{sin}}\left({p}_{0}\left(\cdot \right)\right),{e}^{{p}_{0}\left(\cdot \right)}\mathsf{\text{1}}\right\}\subset {X}^{\left(\gamma ,{U}_{0}\right)}.
Proof. Let P be the family of all continuous k_{ α } seminorms with 0 < k_{ α } ≤ 1. Obviously, the functionals in P satisfy the condition (*) in Theorem 2.2. Moreover, for each p_{ α } ∈ P with k_{ α } ≥ k_{0}, we have
and hence \left\{c{p}_{\alpha}:c\in \mathbb{K},{k}_{\alpha}\ge {k}_{0}\right\}\subset {X}^{\left(\gamma ,{U}_{0}\right)} by Theorem 2.2.
Define f : X → ℝ by f(x) = sin(p_{0}(x)), ∀x ∈ X. For each x ∈ X, u ∈ U_{0} and t ≤ 1, there exists s\in \left[\mid t{\mid}^{{k}_{0}},\mid t{\mid}^{{k}_{0}}\right] and θ ∈ [0,1] such that
i.e.,
where
which implies that f\left(\cdot \right)=sin\left({p}_{0}\left(\cdot \right)\right)\in {X}^{\left(\gamma ,{U}_{0}\right)}.
Define g : X → ℝ by g\left(x\right)={e}^{{p}_{0}\left(x\right)}1, ∀x ∈X. For each x ∈ X, u ∈ U_{0} and t ≤ 1, there exists s\in \left[\mid t{\mid}^{{k}_{0}},\mid t{\mid}^{{k}_{0}}\right] such that
i.e.,
Then, there exists θ,η ∈ [0,1] for which
and
Thus, g\left(\cdot \right)={e}^{{p}_{0}\left(\cdot \right)}1\in {X}^{\left(\gamma ,{U}_{0}\right)}.
Example 2.8 For 0 < p < 1, let L^{p} (0,1) be the space of equivalence classes of measurable functions on [0,1], with
Then (L^{p} (0,1), ·)' = {0} ([[1], p.25]). However, L^{p} (0,1) is locally bounded and hence semiconvex. By Corollary 2.7, if U_{0} = {f : f ≤ 1} and γ(·) = e· ^{p} ∈ C(0), then the demilinear dual{\left({L}^{p}\left(0,1\right),\parallel \cdot \parallel \right)}^{\left(\gamma ,{U}_{0}\right)}contains various nonzero functionals.
A conjecture is that each topological vector space has a nontrivial demilinear dual space. However, this is invalid, even for separable Fréchet space.
Example 2.9 Let \mathcal{M}\left(0,1\right) be the space of equivalence classes of measurable functions on[0,1], with
Then\mathcal{M}\left(0,1\right)is a separable Fréchet space with trivial dual. In fact, the demilinear dual space of\mathcal{M}\left(0,1\right)is also trivial, that is,
Letu\in {\left(\mathcal{M}\left(0,1\right),\parallel \cdot \parallel \right)}^{\left(\gamma ,U\right)}. Let N ∈ ℕ be such that\parallel f\parallel \phantom{\rule{2.77695pt}{0ex}}\le \frac{1}{N}implies f ∈ U and u(f) < 1. Givenf\in \mathcal{M}\left(0,1\right), writef={\sum}_{k=1}^{N}{f}_{k}where f_{ k } = 0 off\left[\frac{k1}{N},\frac{k}{N}\right]. Then\u2225{f}_{k}\u2225\le \frac{1}{N}so
where r_{ i }  1 ≤ γ(1) and s_{ i }  ≤ γ(1) for 2 ≤ I ≤ N. Then
So{sup}_{f\in \mathcal{M}\left(0,1\right)}\mid u\left(f\right)\mid \phantom{\rule{2.77695pt}{0ex}}<+\infty. Since\parallel n{f}_{k}\parallel \phantom{\rule{2.77695pt}{0ex}}\le \frac{1}{N}for each n ∈ ℕ and 1 ≤ k ≤ N, we have {nf_{ k } : n ∈ ℕ, k ∈ ℕ} ⊂ U. Then by Lemma 2.1,
holds for all n ∈ ℕ and 1 ≤ k ≤ N. Letting n → ∞, (7) implies u(f_{ k } ) = 0 for 1 ≤ k ≤ N. Hence, u(f) = 0 by (4). Thus, u = 0.
3 Conclusions on the demilinear dual pair (X, X ^{(γ,U)})
Henceforth, X and Y are topological vector spaces over \mathbb{K}, \mathcal{N}\left(X\right) is the family of neighborhoods of 0 ∈ X, and X^{(γ,U)}is the family of continuous demilinear functionals in {\mathcal{L}}_{\gamma ,U}\left(X,\mathbb{K}\right). Recall that for usual dual pair (X, X') and A ⊂ X, the polar of A, written as A^{°}, is given by
In this article, for the demilinear dual pair (X, X^{(γ,U)}) and A ⊂ X, we denote the polar of A by A^{•}, which is given by
Similarly, for S ⊂ X^{(γ,U)},
Lemma 3.1. Let f\in {\mathcal{L}}_{\gamma ,U}\left(X,Y\right). For every u∈ U and n ∈ ℕ,
Proof. It is similar to the proof of (3)(6) in Example 2.9.
Lemma 3.2. Let S ⊂ X^{(γ,U)}. If S is equicontinuous at 0 ∈ X, then, {S}^{\bullet}\in \mathcal{N}\left(X\right) and sup _{f∈S,x∈B}f(x) < +∞ for every bounded B ⊂ X.
Proof. Assume that S is equicontinuous at 0 ∈ X. There is U\in \mathcal{N}\left(X\right) such that f(x) < 1 for all f ∈ S and x ∈ V. Then V ⊂ S^{•} and hence {S}^{\bullet}\in \mathcal{N}\left(X\right).
Let B ⊂ X be bounded. Since {S}^{\bullet}\cap U\in \mathcal{N}\left(X\right), we have \frac{1}{m}B\subset {S}^{\bullet}\cap U for some m ∈ ℕ. Then for each f ∈ S and x ∈ B,
by Lemma 3.1. Hence, sup _{f∈S,x∈B}f(x) ≤ 2(1 + γ(1))^{m1} 1 < +∞.
Lemma 3.3. Let S ⊂ X^{(γ,U)}. Then S is equicontinuous on X if and only if S is equicontinuous at 0 ∈ X.
Proof. Assume that S is equicontinuous at 0 ∈ X. There is W\in \mathcal{N}\left(X\right) such that f(ω) < 1 for all f ∈ S and ω ∈ W.
Let x ∈ X and ε > 0. By Lemma 3.2, sup _{f ∈S}f(x) = M < +∞. Observing lim _{ t →0 }γ(t) = 0, pick δ ∈ (0, 1) such that \mid \gamma \left(\frac{\delta}{2}\right)\mid <\frac{\epsilon}{2\left(M+1\right)}. By Lemma 2.1, for f ∈ S and u=\frac{\delta}{2}{u}_{0}\in \frac{\delta}{2}\left(W\cap U\right), we have
Thus, f\left[x+\frac{\delta}{2}\left(W+U\right)\right]\subset f\left(x\right)+\left\{z\in \mathbb{K}:\phantom{\rule{2.77695pt}{0ex}}\mid z\mid \phantom{\rule{2.77695pt}{0ex}}<\epsilon \right\} for all f ∈ S, i.e., S is equicontinuous at x.
Theorem 3.4. Let S ⊂ X^{(γ,U)}. Then S is equicontinuous on X if and only if {S}^{\bullet}\in \mathcal{N}\left(X\right).
Proof. If S is equicontinuous, then {S}^{\bullet}\in \mathcal{N}\left(X\right) by Lemma 3.2.
Assume that {S}^{\bullet}\in \mathcal{N}\left(X\right) and ε > 0. Since lim_{ t }_{→0}γ(t) = γ(0) = 0, there is δ > 0 such that γ(t) < ε whenever t < δ. For f ∈ S and x=\frac{\delta}{2}{x}_{0}\in \frac{\delta}{2}\left({S}^{\bullet}\cap U\right), we have f(x_{0}) ≤ 1 and \mid f\left(x\right)\mid \phantom{\rule{2.77695pt}{0ex}}=\mid f\left(\frac{\delta}{2}{x}_{0}\right)\mid \le \mid \gamma \left(\frac{\delta}{2}\right)\mid \mid f\left({x}_{0}\right)\mid \phantom{\rule{2.77695pt}{0ex}}<\epsilon by Lemma 2.1. Thus, f\left[\frac{\delta}{2}\left({S}^{\bullet}\cap U\right)\right]\subset \left\{z\in \mathbb{K}:\phantom{\rule{2.77695pt}{0ex}}\mid z\mid \phantom{\rule{2.77695pt}{0ex}}<\epsilon \right\} for all f ∈ S, i.e., S is equicontinuous at 0 ∈ X. By Lemma 3.3, S is equicontinuous on X.
The following simple fact should be helpful for further discussions.
Example 3.5. Let (L^{p} (0, 1), ·) be as in Example 2.8, U = {f : f  ≤ 1} and γ(t) = e t ^{p} for t\in \mathbb{K} . Then (L^{p} (0, 1), ·)^{(γ,U)}contains nonzero continuous functionals such as ·, sin ·, e^{·}  1, etc. Since (αf)(·) = αf(·) for \alpha \in \mathbb{K} and f∈ (L^{p} (0, 1), ·)^{(γ,U)}, it follows from e^{·}  1 ∈ (L^{p} (0, 1), ·)^{(γ,U)}that \frac{1}{e}\left({e}^{\parallel \cdot \parallel}1\right)\in {\left({L}^{p}\left(0,1\right),\parallel \cdot \parallel \right)}^{\left(\gamma ,U\right)}. If u ∈ U, then u ≤ 1, sin u ≤ u ≤ 1 and \mid \frac{1}{e}\left({e}^{\parallel u\parallel}1\right)\mid \le \frac{e1}{e}<1. Thus, if V is a neighborhood of 0 ∈ L^{p} (0, 1) such that V ⊂ U, then V^{•}contains nonzero functionals such as ·, sin ·, \frac{1}{e}\left({e}^{\parallel \cdot \parallel}1\right), etc.
Corollary 3.6. For every U,V\in \mathcal{N}\left(X\right) and γ∈ C(0), V^{•} = {f ∈ X^{(γ,U)}: f(x) ≤ 1, ∀x ∈ V} is equicontinuous on X.
Proof. Let x ∈ V. Then f(x) ≤ 1, ∀f ∈ V^{•}, i.e., x ∈ (V^{•})^{•}. Thus, V ⊂ (V^{•})^{•} and so {\left({V}^{\bullet}\right)}^{\bullet}\in \mathcal{N}\left(X\right). By Theorem 3.4, V^{•} is equicontinuous on X.
Corollary 3.7. If X is of second category and S ⊂ X^{(γ,U)}is pointwise bounded on X, then {S}^{\bullet}\in \mathcal{N}\left(X\right).
Proof. By Theorem 1.4, S is equicontinuous on X. Then {S}^{\bullet}\in \mathcal{N}\left(X\right) by Theorem 3.4.
Corollary 3.8. Let X be a semiconvex space and S ⊂ X^{(γ,U)}. Then S is equicontinuous on x if and only if there exist finitely many continuous k_{ i }seminorm p_{ i }'s ( 0 < k_{ i } ≤ 1, 1 ≤ i ≤ n < +∞) on x such that
In particular, for a pseminormed space (X, ·) (· is a pseminorm for some p ∈ (0, 1], especially, a norm when p = 1) and S ⊂ X^{(γ,U)}, S is equicontinuous on x if and only if
Proof. Assume that S is equicontinuous. Then {S}^{\bullet}\in \mathcal{N}\left(X\right) by Theorem 3.4. According to Definition 2.6, there exist finitely many continuous k_{ i } seminorm p_{ i } 's (0 < k_{ i } ≤ 1, 1 ≤ i ≤ n < +∞) and ε > 0 such that
Let f ∈ S and p_{ i } (x) ≤ 1, 1 ≤ i ≤ n. Pick n_{0} ∈ ℕ for which {\left(\frac{1}{{n}_{0}}\right)}^{{k}_{0}}<\epsilon, where k_{0} = min_{1≤i≤n}k_{ i } . Then
which implies \frac{x}{{n}_{0}}\in {S}^{\bullet}\cap U and hence \mid f\left(\frac{x}{{n}_{0}}\right)\mid \le 1. By Lemma 3.1,
Thus, {sup}_{f\in S}{sup}_{{p}_{i}\left(x\right)\le 1,1\le i\le n}\mid f\left(x\right)\mid \phantom{\rule{2.77695pt}{0ex}}\le {2\left(1\phantom{\rule{2.77695pt}{0ex}}+\mid \gamma \left(1\right)\mid \right)}^{{n}_{0}1}1<+\infty.
Conversely, suppose that p_{ i } is a continuous k_{ i } seminorm with 0 < k_{ i } ≤ 1 for 1 ≤ i ≤ n < +∞, and (8) holds. Let A=\left\{\frac{1}{M+1}f:f\in S\right\}. Then A ⊂ X^{(γ,U)}and
i.e., {x ∈ X : p_{ i } (x) ≤ 1, 1 ≤ i ≤ n} ⊂ A^{•} and so {A}^{\bullet}\in \mathcal{N}\left(X\right). By Theorem 3.4, A^{•} is equicontinuous on X and S = (1 + M)A is also equicontinuous on X.
Lemma 3.9. Let C\left(X,\mathbb{K}\right)=\left\{f\in {\mathbb{K}}^{X}:f\phantom{\rule{2.77695pt}{0ex}}is\phantom{\rule{2.77695pt}{0ex}}continuous\right\}. For S\subset C\left(X,\mathbb{K}\right), the following (I) and (II) are equivalent.
(I) S is equicontinuous on X.
(II) If(x_{ α } )_{ α }_{∈}I is a net in x such that x_{ α } → x ∈ X, then lim _{ α } f(x_{ α } ) = f(x) uniformly for f ∈ S.
Proof. (I)⇒(II). Let ε > 0 and x_{α} → x in X. Since S is equicontinuous on X, there is W\in \mathcal{N}\left(X\right) such that
Since x_{ α } → x, there is an index α_{0} such that x_{ α }  x ∈ W for all α ≥ α_{0}. Then
Thus, lim _{ α } f(x_{ α } ) = f(x) uniformly for f ∈ S.
(II)⇒(I). Suppose that (II) holds but there exists x ∈ X such that S is not equicontinuous at x.
Then there exists ε > 0 such that for every V\in \mathcal{N}\left(X\right), we can choose f_{ v } ∈ S and z_{ v } ∈ V for which
Since \left(\mathcal{N}\left(X\right),\supset \right) is a directed set, we have {\left(x+{z}_{v}\right)}_{V\in \mathcal{N}\left(X\right)} is a net in X. For every W\in \mathcal{N}\left(X\right),
that is, lim_{ v }(x + z_{ v } ) = x.
By (II), there exists {W}_{0}\in \mathcal{N}\left(X\right) such that f(x + z_{ v } )  f(x) < ε for all f ∈ S and V\in \mathcal{N}\left(X\right) with W_{0} ⊃ V. Then f_{ v } (x + z_{ v } )  f_{ v } (x) < ε for all V\in \mathcal{N}\left(X\right) with W_{0} ⊃ V. This contradicts (9) established above. Therefore, (II) implies (I).
We also need the following generalization of the useful lemma on interchange of limit operations due to E. H. Moore, whose proof is similar to the proof of Moore lemma ([[6], p. 28]).
Lemma 3.10. Let D_{1}and D_{2}be directed sets, and suppose that D_{1} × D_{2}is directed by the relation \left({d}_{1},{d}_{2}\right)\le \left({d}_{1}^{\prime},{d}_{2}^{\prime}\right), which is defined by {d}_{1}\le {d}_{1}^{\prime} and {d}_{2}\le {d}_{2}^{\prime}. Let f: D_{1} × D_{2} → X be a net in the complete topological vector space X. Suppose that:
(a) for each d_{2} ∈ D_{2}, the limitg\left({d}_{2}\right)={lim}_{{D}_{1}}f\left({d}_{1},{d}_{2}\right)exists, and
(b) the limith\left({d}_{1}\right)={lim}_{{D}_{2}}f\left({d}_{1},{d}_{2}\right)exists uniformly on D_{1}.
Then, the three limits
all exist and are equal.
We now establish the AlaogluBourbaki theorem ([[1], p. 130]) for the pair (X, X^{(γ,U)}), where X is an arbitrary nontrivial topological vector space.
Let {\mathbb{K}}^{X} be the family of all scalar functions on X. With the pointwise operations (f + g)(x) = f(x) + g(x) and (t f)(x) = t f(x) for x ∈ X and t\in \mathbb{K}, we have x:{\mathbb{K}}^{X}\to \mathbb{K} is a linear space and each x ∈ X defines a linear functional x:{\mathbb{K}}^{X}\to \mathbb{K} by letting x( f) = f(x) for f\in {\mathbb{K}}^{X}. In fact, for f,g\in {\mathbb{K}}^{X} and \alpha ,\beta \in \mathbb{K},
Then, each x ∈ X produces a vector topology ωx on {\mathbb{K}}^{X} such that
The vector topology V {ωx : x ∈ X} is just the weak * topology in the pair \left(X,{\mathbb{K}}^{X}\right), and f_{ α } → f in \left({\mathbb{K}}^{X},weak*\right) if and only if f_{ α } (x) → f(x) for each x ∈ X ( [[1], p. 12, p. 38]). Note that weak* is a Hausdorff locally convex topology on {\mathbb{K}}^{X}.
Definition 3.11. A subset A ⊂ X^{(γ,U)}is said to be weak * compact in the pair (X, X^{(γ,U)}) or, simply, weak * compact if A is compact in \left({\mathbb{K}}^{X},weak*\right), and A is said to be relatively weak* compact in the pair (X, X^{γ,U} ) or, simply, relatively weak* compact if in \left({\mathbb{K}}^{X},weak*\right) the closure \u0100 is compact and \u0100\subset {X}^{\left(\gamma ,U\right)}.
For A ⊂ X^{(γ,U)}, {\u0100}^{weak*} stands for the closure of A in \left({\mathbb{K}}^{X},weak*\right).
Theorem 3.12. For every V\in \mathcal{N}\left(X\right), V^{•} = {f ∈ X^{(γ,U)}: f(x) ≤ 1, ∀x ∈ V} is weak* compact in the pair (X, X^{(γ,U)}), and every equicontinuous S ⊂ X^{(γ,U)}is relatively weak* compact in the pair (X, X^{(γ,U)}).
Proof. For each x ∈ X, let x(f) = f(x) for f ∈ {\mathbb{K}}^{X}, then x:{\mathbb{K}}^{X}\to \mathbb{K} is a linear functional. Let V\in \mathcal{N}\left(X\right). By Corollary 3.6, V^{•} is equicontinuous on X and, by Lemma 3.2, x(V^{•}) = {f(x): f ∈ V^{•}} is bounded in \mathbb{K} for each x ∈ X, i.e., for each x ∈ X, x(V^{•}) is totally bounded in \mathbb{K} and so V^{•} is totally bounded in \left({\mathbb{K}}^{X},\omega x\right) for each x ∈ X ( [[1], p. 84, Theorem 6]. But the weak* topology for {\mathbb{K}}^{X} is just V {ωx : x ∈ X} and so V^{•} is totally bounded in \left({\mathbb{K}}^{X},weak*\right) ([[1], p. 85, Theorem 7].
Let (f_{ α } )_{α∈I}⊂ V^{•} be a Cauchy net in \left({\mathbb{K}}^{X},weak*\right). Then lim_{ α } f_{ α }(x) = f(x) exists at each x ∈ X and so f_{ α } → f in \left({\mathbb{K}}^{X},weak*\right). For x ∈ X, u ∈ U and t\in \left\{z\in \mathbb{K}:\phantom{\rule{2.77695pt}{0ex}}\mid z\mid \phantom{\rule{2.77695pt}{0ex}}\le 1\right\},
By passing to a subnet if necessary, we assume that r_{ α } → r and s_{ α } → s in \mathbb{K}. Then r  1 = lim _{ α } r_{ α }  1 ≤ γ(t), s = lim _{ α } s_{ α }  ≤ γ(t) and
This shows that f\in {\mathcal{L}}_{\gamma ,U}\left(X,\mathbb{K}\right).
Let x_{ β } → x in X. Since V^{•} is equicontinuous on X and f_{ α } ∈ V^{•} for all α ∈ I, it follows from Lemma 3.9 that lim_{ β } f_{ α }(x_{ β } ) = f_{ α } (x) uniformly for α ∈ I. Then
by Lemma 3.10, i.e., f:X\to \mathbb{K} is continuous and hence f ∈ X^{(γ,U)}. Moreover, f(x) = lim _{ α } f_{ α } (x) ≤ 1 for each x ∈ V, i.e., f ∈ V^{•}. Thus, V^{•} is complete in \left({\mathbb{K}}^{X},weak*\right). Since \left({\mathbb{K}}^{X},weak*\right) is a topological vector space and V^{•} is both totally bounded and complete in \left({\mathbb{K}}^{X},weak*\right), we have V^{•} is compact in \left({\mathbb{K}}^{X},weak*\right), i.e., V^{•} is weak* compact in the pair (X, X^{(γ,U)}) ( [[1], p. 88, Theorem 7]).
Assume that S ⊂ X^{(γ,U)}is equicontinuous on X. By Lemma 3.2,
{S}^{\bullet}=\left\{x\in X:\phantom{\rule{2.77695pt}{0ex}}\mid f\left(x\right)\mid \phantom{\rule{2.77695pt}{0ex}}\le 1,\forall f\in S\right\}\in \mathcal{N}\left(X\right), it follows from what is established above that (S^{•})^{•} = {f ∈ X^{(γ,U)}: f(x) ≤ 1, ∀x ∈ S^{•}} is compact in the Hausdorff space \left({\mathbb{K}}^{X},weak*\right). Then S ⊂ (S^{•})^{•} shows that {\stackrel{\u0304}{S}}^{weak*}\subset {\left({S}^{\bullet}\right)}^{\bullet}\subset {X}^{\left(\gamma ,U\right)} and S is relatively weak* compact in (X, X^{(γ,U)}).
Theorem 3.12 is a version of AlaogluBourbaki theorem for the demilinear dual pair (X, X^{(γ,U)}), by which we can establish an improved BanachAlaoglu theorem ( [[1], p. 130] as follows.
Corollary 3.13 (BanachAlaoglu). Let X be a seminormed space and M > 0. Then
is weak* compact in the pair (X, X^{(γ,U)}).
Proof. Since sup_{f∈S}sup_{x≤1}f(x) ≤ M < +∞, Corollary 3.8 shows that S is equicontinuous on X. By Theorem 3.12, {\stackrel{\u0304}{S}}^{weak*}\subset {X}^{\left(\gamma ,U\right)} and {\stackrel{\u0304}{S}}^{weak*} is compact in \left({\mathbb{K}}^{X},weak*\right).
Let (f_{ α } )_{α∈I}be a net in S such that lim_{ α } f_{ α }(x) = f(x) at each x ∈ X. Then f ∈ X,^{(γ,U)}and
i.e., f ∈ S. Thus, {\stackrel{\u0304}{S}}^{weak*}=S.
Theorem 3.14. Let X be a separable space, K a weak* compact set in X^{(γ,U)}, S an equicontinuous set in X^{(γ,U)}, and V\in \mathcal{N}\left(X\right), {V}^{\bullet}=\left\{f\in {X}^{\left(\gamma ,U\right)}:\mid f\left(x\right)\mid \phantom{\rule{2.77695pt}{0ex}}\le 1,\forall x\in V\right\}. Then(S, weak*) is metrizable, and both (K, weak*) and (V^{•}, weak*) are compact metric spaces.
Proof. Assume that {\left\{{x}_{n}\right\}}_{n=1}^{\infty} is dense in X. Let
Then, d(·,·) is a pseudometric on {\mathbb{K}}^{X}. If f, g ∈ X^{(γ,U)}and d(f, g) = 0, then f(x_{ n } ) = g(x_{ n } ) for all n. Since both f and g are continuous on X and {\left\{{x}_{n}\right\}}_{n=1}^{\infty} is dense in X, f(x) = g(x) for all x ∈ X, i.e., f = g. This shows that (X^{(γ,U)}, d) is a metric space, and f_{ k } → f in (X^{(γ,U)}, d) if and only if lim_{ k } f_{ k }(x_{ n } ) = f(x_{ n } ) for each n ∈ ℕ. Hence, weak* is stronger than d(·, ·) and so the compact space (K, weak*) is homeomorphic to the (Hausdorff) metric space (K, d). Thus, (K, weak*) is a compact metric space.
By Theorem 3.12, in \left({\mathbb{K}}^{X},weak*\right) the closure {\stackrel{\u0304}{S}}^{weak*}\subset {X}^{\left(\gamma ,U\right)}, and both \left({\stackrel{\u0304}{S}}^{weak*},weak*\right) and (V^{•}, weak*) are compact and so they are compact metric spaces.
The following special case of Theorem 3.14 is a wellknown fact ([[1], p. 143]).
Corollary 3.15. Let X be a separable locally convex space with the dual X', K a weak* compact set in X', S an equicontinuous set in X', and V\in \mathcal{N}\left(X\right), V^{°} = {f ∈ X^{0} : f(x) ≤ 1, ∀x ∈ V}. Then (S, weak*) is metrizable, and both (K, weak*) and (V^{°}, weak*) are compact metric spaces.
Corollary 3.16. Let X be a separable space and S an equicontinuous set in X^{(γ,U)}. Every sequence {f_{ n } } in S has a subsequence \left\{{f}_{{n}_{k}}\right\} such that {lim}_{k}{f}_{{n}_{k}}\left(x\right)=f\left(x\right) exists at each X∈ X and the limit function f ∈ X^{(γ,U)}, i.e., f is both continuous and demilinear.
Proof. By Theorems 3.12 and 3.14, {\stackrel{\u0304}{S}}^{weak*}\subset {X}^{\left(\gamma ,U\right)} and \left({\stackrel{\u0304}{S}}^{weak*},weak*\right) is a compact metric space. Then \left({\stackrel{\u0304}{S}}^{weak*},weak*\right) is sequentially compact.
Combining Theorem 1.4 and Corollary 3.16, we have the following
Corollary 3.17. Assume that X is of second category and separable, e.g., separable Fréchet spaces such as L^{p} (0, 1)(p > 0), C[0,1], c_{0}, c, l^{p}(p > 0), etc. If S ⊂ X^{(γ,U)}is pointwise bounded on X, then every sequence {f_{ n } } in S has a subsequence \left\{{f}_{{n}_{k}}\right\} such that {lim}_{k}{f}_{{n}_{k}}\left(x\right)=f\left(x\right) exists at each x∈ X, and f ∈ X^{(γ,U)}.
For C ≥ 1 and δ > 0, letting γ(t) = Ct for t ∈ ℝ and U = (δ,δ), we have γ ∈ C(0) and U\in \mathcal{N}\left(\mathbb{R}\right). Then let ℝ^{(C,δ)}= ℝ^{(γ,U)}. It is easy to see that every f\in {\mathcal{L}}_{\gamma ,U}\left(\mathbb{R},\mathbb{R}\right) is continuous and so {\mathbb{R}}^{\left(C,\delta \right)}={\mathcal{L}}_{\gamma ,U}\left(\mathbb{R},\mathbb{R}\right). Thus, ℝ^{(C,δ)}contains all linear functions and various nonlinear functions. It is noted that many functions in ℝ^{(C,δ)}have very complicated graphs.
For S ⊂ ℝ^{(C,δ)}, there is an interesting fact: a local behavior in a small interval (ε, ε) implies a nice behavior on (∞, +∞).
Example 3.18. Let S ⊂ ℝ^{(C,δ)}. If there exists M, ε > 0 such that f(x) ≤ M for every f ∈ S and x ∈ (ε, ε), then every {f_{ n } } ⊂ S has a subsequence \left\{{f}_{{n}_{\mathbb{K}}}\right\} such that {lim}_{k}{f}_{{n}_{k}}\left(x\right)=f\left(x\right) exists at each x∈ ℝ, and f ∈ ℝ^{(C,δ)}.
In fact, ℝ is separable and \left(\epsilon ,\epsilon \right)\in \mathcal{N}\left(\mathbb{R}\right). The assumption shows that
By Theorem 3.14, ((ε, ε)^{•}, weak*) is a compact metric space and so it is sequentially compact. Similarly, we have
Example 3.19. Let p > 0 and S ⊂ (L^{p} (0, 1))^{(γ,U)}. If there exists ε > 0 such that f(x) ≤ 1 whenever f ∈ S and X ∈ L^{p} (0, 1) with x < ε, then every {f_{ n } } ⊂ S has a subsequence \left\{{f}_{{n}_{k}}\right\} such that {lim}_{k}{f}_{{n}_{k}}\left(x\right)=f\left(x\right) exists for all x∈ L^{p} (0, 1), and f ∈ (L^{p} (0, 1))^{(γ,U)}.
We shall show that the condition "sup _{f∈S,x<ε} f(x) ≤ 1" in Example 3.19 can be weakened as "sup_{f∈S}f(x) < +∞, ∀ x < ε" (see Corollary 3.20).
In general, combining Theorems 3.12 and 3.14, we have
Corollary 3.20. Let S ⊂ X^{(γ,U)}. If there exists V\in \mathcal{N}\left(X\right) such that sup _{f∈S,x∈V}f(x) < +∞, then
(a) S is equicontinuous on X,
(b) S is relatively weak * compact,
(c) every net(f_{ α } ) in S has a subnet (f_{ ξ } (α)) such that lim_{ξ(α)}f_{ ξ } (α)(x) = f(x) exists for all x ∈ X, and f ∈ X^{(γ,U)}.
If, in addition, x is separable, then
(d) every {f_{ n } } ⊂ S has a subsequence \left\{{f}_{{n}_{k}}\right\} such that {lim}_{k}{f}_{{n}_{k}}\left(x\right)=f\left(x\right) exists for all x∈ X, and f ∈ X^{(γ,U)}.
In fact, for M = sup_{f∈S,x∈V}f(x), we have A=\left\{\frac{1}{M+1}f:f\in S\right\}\subset {V}^{\bullet} and (a)(d) hold for A, i.e., S satisfies (a)(d).
If X is of second category, then the condition "there exists V\in \mathcal{N}\left(X\right) such that sup_{f∈S,x∈V}f(x) < +∞" in Corollary 3.20 can be weakened as "there exists V\in \mathcal{N}\left(X\right) such that sup f∈S f(x) < +∞, ∀x ∈ V".
To see this, we first establish a simple fact.
Lemma 3.21. Let \mathrm{\Gamma}\subset {\mathcal{L}}_{\gamma ,U}\left(X,Y\right). If there exists V\in \mathcal{N}\left(X\right) such that{f(x): f ∈ Г} is bounded at each x ∈ V, then {f(x): f ∈ Г} is bounded at each x ∈ X.
Proof. Let x ∈ X. There exists n_{0} ∈ ℕ such that \frac{1}{{n}_{0}}x\in V\cap U. By Lemma 3.1, for each f ∈ Г, we have
Then
Since \frac{x}{{n}_{0}}\in V, \left\{f\left(\frac{x}{{n}_{0}}\right):f\in \mathrm{\Gamma}\right\} is bounded and so \left\{tf\left(\frac{x}{{n}_{0}}\right):f\in \mathrm{\Gamma},\mid t\mid \phantom{\rule{2.77695pt}{0ex}}\le {2\left(1\phantom{\rule{2.77695pt}{0ex}}+\mid \gamma \left(1\right)\mid \right)}^{{n}_{0}1}1\right\} is bounded.
Now we can improve Theorems 1.4 and 1.5 as follows.
Theorem 3.22. Assume that x is of second category and \mathrm{\Gamma}\subset \left\{f\in {\mathcal{L}}_{\gamma ,U}\left(X,Y\right):f\phantom{\rule{2.77695pt}{0ex}}iscontinuous\right\}. If there exists V\in \mathcal{N}\left(X\right) such that Г is pointwise bounded on V, then Г is equicontinuous on X, and Г is uniformly bounded on each bounded subset of X.
Corollary 3.23. Assume that x is of second category and S ⊂ X^{(γ,U)}. If there exists V\in \mathcal{N}\left(X\right) such that sup_{f∈S}f(x) < +∞ at each x ∈ V, then (a)(c) hold for S. If, in addition, X is separable, then (d) holds for S.
We now show that every equicontinuous S ⊂ X^{(γ,U)}has a nice behavior on any compact subset of X.
Theorem 3.24. Let X be a Hausdorff topological vector space. If S is an equicontinuous subset of X^{(γ,U)}and Ω is a compact subset of X, then every {f_{ n } } ⊂ S has a subsequence \left\{{f}_{{n}_{k}}\right\} such that {lim}_{k}{f}_{{n}_{k}}\left(x\right)=f\left(x\right) uniformly for x∈ Ω and f : f:\mathrm{\Omega}\to \mathbb{K}is continuous.
Proof. Let K = {f _{Ω}: f ∈ S}. Then K ⊂ C(Ω) and K is equicontinuous at each x ∈ Ω. Suppose that sup _{f∈K}f_{∞} = sup _{f∈K,x∈Ω}f(x) = +∞. Then there exist sequences {f_{ n } } ⊂ S and {x_{ n } } ⊂ Ω such that f_{ n } (x_{ n } ) > n, ∀n ∈ ℕ. By Lemma 3.2, we may assume that x_{ n } ≠ x_{ m } for n ≠ m.
Since Ω is compact, {\left\{{x}_{n}\right\}}_{n=1}^{\infty} has a cluster point x ∈ Ω.
Since S is equicontinuous at x, there exists V\in \mathcal{N}\left(X\right) such that f(y)  f(x) < 1 for all f ∈ S and y ∈ x + V, i.e., f(y) < f(x) + 1 for all f ∈ S and y ∈ x + V. Observing that f_{ n } (x_{ n } ) > n for all n ∈ ℕ and {f_{ n } } ⊂ S, there exists n_{0} ∈ ℕ such that x_{ n } ∉ x + V for all n > n_{0}. Since (x + V) ∩ Ω contains some x_{ n } with x_{ n } ≠ x, it follows that
where m ≤ n_{0}. But X is Hausdorff, so Ω is also Hausdorff. Then there exists {V}_{0}\in \mathcal{N}\left(X\right) such that V_{0} ⊊ V and (x + V_{0}) ∩ (Ω ∩ {y_{1}, y_{2}, · · ·, y_{ m } }) = ∅. Hence x_{ n } ∈ (x + V_{0}) ∩ Ω implies that x_{ n } = x.
This contradicts the fact that x is a cluster point of {\left\{{x}_{n}\right\}}_{n=1}^{\infty}. Hence, {sup}_{f\in \mathbb{K}}\parallel f\mid {\mid}_{\infty}<+\infty
By the ArzelaAscoli theorem, K is relatively compact in the metric space (C(Ω), ·_{∞}). Hence, every {f_{ n } } ⊂ S has a subsequence \left\{{f}_{{n}_{k}}\right\} such that \parallel {{f}_{{n}_{K}}\mid}_{\mathrm{\Omega}}f{\parallel}_{\infty}\to 0, where f ∈ C(Ω), i.e., {lim}_{k}{f}_{{n}_{k}}\left(x\right)=f\left(x\right) uniformly for x ∈ Ω.
Corollary 3.25. Let X = ℝ ^{n} or ℂ ^{n}, ε > 0 and D_{ m } = {x ∈ X : x ≤ mε}, ∀m ∈ ℕ. If S ⊂ X^{(γ,U)}is pointwise bounded on D_{1}, then every sequence {f_{ k } } ⊂ S has a subsequence \left\{{f}_{{k}_{i}}\right\} such that {lim}_{i}{f}_{{k}_{i}}\left(x\right)=f\left(x\right) uniformly on each D_{ m }, where f ∈ X^{(γ,U)}.
Proof. Theorem 3.22 shows that S is equicontinuous on X and, by Theorem 3.24, {f_{ k } } has a subsequence \left\{{f}_{{\mathbb{K}}_{i}}\right\} such that {lim}_{i}{f}_{{k}_{i}}\left(x\right) exists uniformly on D_{1}. Then {\left\{{f}_{{k}_{i}}\right\}}_{i=2}^{\infty} has a subsequence \left\{{f}_{{k}_{{i}_{v}}}\right\} such {lim}_{v}{f}_{{k}_{{i}_{v}}}\left(x\right) exists uniformly on D_{2}. Proceeding inductively, the diagonal procedure yields a subsequence {g_{ i } } of {f_{ k } } such that lim_{ i } g_{ i }(x) exists uniformly on each D_{ m } . Then lim_{ i } g_{ i }(x) = f(x) exists at each x ∈ X and f\in {\stackrel{\u0304}{S}}^{weak*} in \left({\mathbb{K}}^{X},weak*\right). By Theorem 3.12, f ∈ X^{(γ,U)}.
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4 Competing interests
The authors declare that they have no competing interests.
5 Authors' contributions
RL gave the basic ideas and composed the main skeleton of this paper. His work includes the main theorems in section 2 and 3, and some concrete examples. AC provided more examples in section 2, proved some corollaries in section 2, 3, and drafted the manuscript. SZ participated in the discussion of the ideas and provided some insightful suggestion. All authors read and approved the final manuscript.
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Li, R., Chen, A. & Zhong, S. Demilinear duality. J Inequal Appl 2011, 128 (2011). https://doi.org/10.1186/1029242X2011128
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DOI: https://doi.org/10.1186/1029242X2011128