Nuclear mapping and a Riemann approach to vector valued integration

Shizu Nakanishi

Abstract

The purpose of this paper is to show a natural setting under which we consider vector valued functions integrable in the generalized Riemann sense, using nuclear mappings. The results of this paper contain all of the results published already in [17-22].

K. Kunugi introduced the method of ranked space as a method for the mathematical analysis in 1954 (cf. [6]). Following this, we introduced the notion of ranked union space, and showed that the spaces S, S′, D, D′, etc., occurring in the distribution theory of L. Schwartz [24], can be treated as ranked union spaces called “Ranked (UCs-N) Spaces” in short, without changing the notion of convergence for the countable sequence of points ([9], [11]). As other typical ranked (UCs-N) spaces we know the spaces C([0, 1], S), C([0, ∞), S), C([0, 1], D), C([0, ∞), D), C([0, 1], S′), C([0, ∞), S′), C([0, 1], D′), C([0, ∞), D′), etc. ([13]). This paper is concerned with the McShane and Henstock-Kurzweil integrals. We tried a Riemann approach to integration for the functions taking values in a ranked Hilbertian (UCs-N) space endowed with the nuclearity, and knew that the ranked Hilbertian (UCs-N) space endowed with the nuclearity is a natural setting under which we consider the vector valued integration in the generalized Riemann sense. In this paper, we will explain the results obtained without use of the notion of ranked space. Because, we do not necessarily need the theory of ranked space on the study of the Riemann type integration. We refer to the book [1] for the terminology concerning Hilbert space etc.

I. (UCs-N) spaces

1. (UCs-N) spaces. Let X be a vector space over the real numbers, and let (X_{α}, {p_{m}^{α}}_{m=0}^{∞}) (α ∈ Σ) be a family of subspaces X_{α} of X such that a sequence of semi-norms {p_{m}^{α}}_{m=0}^{∞} is defined on X_{α} for each α. Suppose that they satisfy the following conditions (I)-(V).

(I) ∪_{α∈Σ}X_{α} = X.

(II) Σ is an upward directed set with the ordering ≤ (see [27, p. 103] for the definition).

(III) α ≤ β if and only if X_{α} ⊂ X_{β}.

(IV) For each α ∈ Σ, p_{0}^{α}(x) ≤ p_{1}^{α}(x) ≤ … for x ∈ X.

(V) If α ≤ β, then p_{m}^{α}(x) ≥ p_{m}^{β}(x), for x ∈ X_{α} and m = 0, 1, 2, . . ..

Convergence, Cauchy sequence, Separation axiom

In the space X mentioned in the above, the notion concerned with “convergence” is defined only for the countable sequence of points as follows.

(C_{1}) A sequence x_{i} (i = 1, 2, …) is said to be convergent to x in X if and only if there exists an α such that x_{i} (i = 1, 2, …) and x are contained in X_{α} and the sequence is convergent to x in the space X_{α} topologized by {p_{m}^{α}}_{m=0}^{∞} (cf. [2, p. 15]).

(C_{2}) A sequence x_{i} (i = 1, 2, …) is said to be a Cauchy sequence in X if and only if there exists an α such that x_{i} (i = 1, 2, …) are contained in X_{α} and the sequence is a Cauchy sequence in the space X_{α} topologized by {p_{m}^{α}}_{m=0}^{∞}.

(C_{3}) The space X is said to be separated if x = y whenever lim x_{i} = x and lim x_{i} = y.

By (C_{1}) and (C_{2}), we see that the space X is separated if and only if for every α ∈ Σ, the space X_{α} topologized by {p_{m}^{α}}_{m=0}^{∞} is separated.

Definition 1. If X is a vector space over the real numbers endowed with (X_{α}, {p_{m}^{α}}_{m=0}^{∞}) (α ∈ Σ) satisfying (I)-(V) and if, on X, convergence, Cauchy sequence and separation axiom are defined by (C_{1}), (C_{2}) and (C_{3}), respectively, then the space X is called a (UCs-N) space defined by (X_{α}, {p_{m}^{α}}_{m=0}^{∞}) (α ∈ Σ) ((UCs-N ) standing for “union of countably semi-normed spaces”), and each space (X_{α}, {p_{m}^{α}}_{m=0}^{∞}) is called a component space of the (UCs-N) space X. We sometimes use the same notation (X_{α}, {p_{m}^{α}}_{m=0}^{∞}) to denote the space X_{α} topologized by {p_{m}^{α}}_{m=0}^{∞}.

In particular:

[1] When Σ is a set consisting of a single element, say α, and p_{0}^{α}(x) = p_{1}^{α}(x) = ⋯, the space X is called a (s-N) space. In particular, if the common semi-norm is a norm, the space X is called a (N) space. We usually denote such a (s-N) space X by (X, p) as a vector space endowed with a semi-norm p.

[2] When Σ is a set consisting of a single element, say α, and p_{0}^{α}(x) ≤ p_{1}^{α}(x) ≤ ⋯, the space X is called a (Cs-N) space. In particular, if p_{m}^{α} is a norm for each m, then X is called a (CN) space. We usually denote such a (Cs-N) space X by (X, {p_{m}}_{m=0}^{∞}).

[3] When p_{0}^{α}(x) = p_{1}^{α}(x) = ⋯ for each α ∈ Σ, the space X is called a (Us-N) space. In particular, if the common semi-norm is a norm, the space X is called a (UN) space.

[4] When the space X is a (Us-N) space (resp. (UN) space) such that Σ is a countable set, the space X is called a (CUs-N) space (resp. (CUN) space).

[5] When, for each α ∈ Σ, p_{m}^{α} is a norm for m = 0, 1, 2, …, the (UCs-N) space X is called a (UCN) space.

[6] When the space X is a (UCs-N) space (resp. (UCN) space) such that Σ is a countable set, the space X is called a (CUCs-N) space (resp. (CUCN) space).

Example 1

The spaces S, S′, D and D′ can be treated as follows without changing the meaning of convergence for a countable sequence of points.

(1) S: a complete separated (CN) space ([10, (2), p. 198], [11, Proposition 1 p. 355])

(2) S′: a separated (CUN) space with complete component spaces ([10, II, p. 806], [11, Theorem 1, p. 359])

(3) D: a separated (CUCN) space with complete component spaces ([10, p. 195]. [11, Propositions 2 and 4, pp. 356, 357])

(4) D′: a separated (UCs-N) space with complete component spaces ([11, Theorem 2, p. 364], [11, Propositions 2 and 4, pp. 356, 357])

2. (UCs-N) space valued continuous functions. Let X be a (UCs-N) space with component spaces (X_{α}, {p_{m}^{α}}_{m=0}^{∞}) (α ∈ Σ). An X-valued function f defined on [a, b] is said to be continuous at t ∈ [a, b] if there are an α ∈ Σ and a neighborhood V (t) of t in [a, b] such that

(1) the image of V (t) by f is contained in X_{α},

(2) f is continuous at t as a mapping from V (t) into the component space (X_{α}, {p_{m}^{α}}_{m=0}^{∞}), more precisely, if lim t_{i} = t in V (t), then lim f(t_{i}) = f(t) in the countably semi-normed space X_{α} topologized by {p_{m}^{α}}_{m=0}^{∞}.

An X-valued function f is said to be continuous on [a, b] if it is continuous at every point of [a, b].

In general, we do not know whether or not t he following statement (UCs-N,c) is true:

(UCs-N,c) If a (UCs-N) space valued function f is continuous on [a, b], then there exists an α such that the image of [a, b] by f is contained in X_{α} and f is continuous on [a, b] as a function taking values in the space X_{α} topologized by {p_{m}^{α}}_{m=0}^{∞}.

We have characterized (UCs-N) spaces for which the statement (UCs-N,c) is true (see [14, Propositions 15 and 16, pp. 116, 117]). In particular, we have:

Example 2

In the S, S′, D and D′-valued cases, the statement (UCs-N,c) is true ([14, p. 117]).

3. (UCs-N) space valued countably additive measures. Let X be a separated (UCs-N) space with component spaces (X_{α}, {p_{m}^{α}}_{m=0}^{∞}) (α ∈ Σ) and M a ring of sets. A function F : M → X is called a countably additive vector measure if F (∅) = 0 and F (∪_{i=1}^{∞}A_{i}) = lim_{n→∞} ∑_{i=1}^{n} F (A_{i}) for every sequence {A_{i}} of mutually disjoint sets of M such that ∪_{i=1}^{∞}A_{i} ∈ M (cf. [12, p. 807]). In general, we do not know whether or not the following statement (UCs-N,m) is true.

(UCs-N,m) Let M be a σ-ring of sets and let F : M → X be a countably additive vector measure. Then, there exists an α such that the image of [a, b] by F is contained in X_{α} and F is countably additive as a function taking values in the space X_{α} topologized by {p_{m}^{α}}_{m=0}^{∞}.

It is pointed out that the statement (UCs-N,m) is true if the (UCs-N) space has the property (†) indicated in [12, p. 807] (see [12, Theorem 2, p. 808]). In particular, we have:

Example 3

In the S, S′, D and D′-valued cases, the statement (UCs-N,m) is true ([12, p. 808]).

The statements in Sections 2 and 3 motivate us to define Bochner, Mc-Shane and Henstock-Kurzweil integrals as in Definitions 6 and 7 of Chapter IV below for the functions taking values in the (UCs-N) spaces.

II. Nuclear Hilbertian (UCs-N) spaces

@Definition 2. Let X_{α} and X_{β} be Hilbert spaces with scalar products ( , )_{α} and ( , )_{β}, and p_{α} and p_{β} the norms associated with ( , )_{α} and ( , )_{β}, respectively. A continuous linear mapping T_{β}^{α} of the Hilbert space X_{α} into the Hilbert space X_{β} is called a nuclear mapping if it has the following form.

N(α, β) : T_{β}^{α}(x) = ∑_{k=1}^{∞} λ_{βk}^{α}(x, e_{k}^{α})_{α}e_{k}^{β} for x ∈ X_{α},

where {e_{k}^{α}}_{k=1}^{∞} and {e_{k}^{β}}_{k=1}^{∞} are orthonormal systems of vectors in X_{α} and X_{β}, respectively, λ_{βk}^{α} > 0 and ∑_{k=1}^{∞} λ_{βk}^{α} < ∞.

Next, we define a (UCs-N) space endowed with nuclearity. We begin by defining a mapping.

Let X be a separated (UCs-N) space with complete component spaces (X_{α}, {p_{m}^{α}}_{m=0}^{∞}) (α ∈ Σ) such that, on each X_{α}, for m = 0, 1, 2, … a positive hermitian form ( , )_{m}^{α} is defined and p_{m}^{α} is the semi-norm associated with ( , )_{m}^{α}.

Let α ≤ β and m ≥ n, and let T_{β}^{α} be the embedding mapping of X_{α} into X_{β}, i.e., T_{β}^{α}(x) = x for x ∈ X_{α}. Since then X is a (UCs-N) space, we have X_{α} ⊂ X_{β} and p_{m}^{α}(x) ≥ p_{n}^{β}(x) for x ∈ X_{α}. Therefore

p_{m}^{α}(x) ≥ p_{n}^{β}(T_{β}^{α}(x)) for x ∈ X_{α}.

Now, consider the quotient spaces X_{α}/N(α, m) and X_{β}/N(β, n), where N(α,

m) = {x ∈ X_{α} : p_{m}^{α}(x) = 0} and N(β, n) = {x ∈ X_{β} : p_{n}^{β}(x) = 0}. Then, the quotient spaces X/N(α, m) and X/N(β, n), denoted by X_{m}^{α} and X_{n}^{β}, become prehilbert spaces with non-degenerate positive hermitian forms, still denoted by ( , )_{m}^{α} and ( , )_{n}^{β}. We denote the norm associated with ( , )_{m}^{α} (resp. ( , )_{n}^{β}) by p_{m}^{α} (resp. p_{n}^{β}) and the element of X_{m}^{α} (resp. X_{n}^{β}) having x ∈ X_{α} (resp. X_{β}) as a representative by [x]_{m}^{α} (resp. [x]_{n}^{β}). Let T_{βn}^{αm} be the mapping of the prehilbert space X_{m}^{α} into the prehilbert space X_{n}^{β} defined as follows.

T_{βn}^{αn}([x]_{m}^{α}) = [x]_{n}^{β}.

Then

p_{m}^{α}([x]_{m}^{α}) ≥ p_{n}^{β}(T_{βn}^{αm}([x]_{m}^{α})) for x_{m}^{α} ∈ X_{m}^{α},

and the mapping T_{βn}^{αm} is a continuous linear mapping of X_{m}^{α} into X_{n}^{β}. Next, we denote the completions of prehilbert spaces X_{m}^{α} and X_{n}^{β} with respect to p_{m}^{α} and p_{n}^{β} by X̂_{m}^{α} and X̂_{n}^{β}, respectively. Then, they become Hilbert spaces. If {[x_{i}]_{m}^{α}}_{i=1}^{∞} is a Cauchy sequence in X_{m}^{α}, then {[x_{i}]_{n}^{β}}_{i=1}^{∞} is a Cauchy sequence in X_{n}^{β}. Hence, the element of X̂_{n}^{β} having the Cauchy sequence {[x_{i}]_{n}^{β}}_{i=1}^{∞} as a representative is uniquely determined by the element of X̂_{m}^{α} having the Cauchy sequence {[x_{i}]_{m}^{α}}_{i=1}^{∞} as a representative. We denote the correspondence by T̂_{βn}^{αm}. Then, T̂_{βn}^{αm} is a continuous linear mapping of X̂_{m}^{α} into X̂_{n}^{β} such that

p̂_{m}^{α}(x̂_{m}^{α}) ≥ p̂_{n}^{β}(T̂_{βn}^{αm}(x̂_{m}^{α})) for x̂_{m}^{α} ∈ X̂_{m}^{α},

where p̂_{m}^{α} and p̂_{n}^{β} are the norms associated with the scalar products on X̂_{m}^{α} and X̂_{n}^{β}, respectively.

Definition 3. Let X be a separated (UCs-N) space with complete component spaces (X_{α}, {p_{m}^{α}}_{m=0}^{∞}) (α ∈ Σ) such that, on each X_{α}, for m = 0, 1, 2, …, a positive hermitian form ( , )_{m}^{α} is defined and p_{m}^{α} is the semi-norm associated with ( , )_{m}^{α}. Suppose that, corresponding to each α ∈ Σ, we can find

(†) a β ∈ Σ and increasing sequences of non-negative integers {m(0) < m(1) < ⋯} and {n(0) < n(1) < ⋯} such that:

(1) α ≤ β,

(2) m(i) ≥ n(i) for i = 0, 1, 2, …, and

(3) T̂_{βn(i)}^{αm(i)} is nuclear for i = 0, 1, 2, …, where T̂_{βn(i)}^{αm(i)} is the continuous linear mapping of X̂_{m(i)}^{α} into X̂_{n(i)}^{β} defined in the above. Then we call such a space X a nuclear Hilbertian (UCs-N) space. β, {m(i)}, {n(i)} is a system associated with α. We have:

Proposition 1. If m ≥ m(i) and n(i) ≥ n for some i, then T̂_{βn}^{αm} is a nuclear mapping of X̂_{m}^{α} into X̂_{n}^{β}.

Example 4

(a) Nuclear Hilbertian (CN) space. Let X be a vector space endowed with a sequence of norms {p_{m}}_{m=0}^{∞}. Suppose that the sequence {p_{m}}_{m=0}^{∞} is non-decreasing and all of the norms p_{m} are compatible (see [2, p. 13] for the definition). Completing the space X by p_{m} , we obtain a Banach space. We denote the Banach space by X̂_{m}. In this case, the space X̂_{m} (m = 0, 1, 2, …) can be considered to have the following relationship with each other and with X.

X̂_{0} ⊃ X̂_{1} ⊃ ⋯ ⊃ X.

The topological space X topologized by such a {p_{m}}_{m=0}^{∞} has been called a countably normed space [2, p. 15]. If the space is complete, then we have X = ∩_{m=0}^{∞}X̂_{m} ([2, p. 17]).

Now, consider a vector space X endowed with a sequence of nondegenerate positive hermitian forms ( , )_{m} (m = 0, 1, 2, …). Let p_{m} be the norm associated with ( , )_{m}. In this case, the space X̂_{m} defined as in the above becomes a Hilbert space. Suppose that the sequence {p_{m}}_{m=0}^{∞} is nondecreasing and compatible (so the space (X, {p_{m}}_{m=0}^{∞}) is a countably normed space) and the space (X, {p_{m}}_{m=0}^{∞}) is complete. Then, the countably normed space (X, {p_{m}}_{m=0}^{∞}) has been called a nuclear countably Hilbert space if for m > n, the embedding T̂_{n}^{m} of the Hilbert space X̂_{m} to the Hilbert space X̂_{n} is a nuclear mapping [3, p. 62]. A nuclear countably Hilbert space is an example of vector space which can be considered as a nuclear Hilbertian (CN) space. The space S is such a typical space ([18, p. 331]).

(b) Nuclear Hilbertian (CUCN) space such that each component space is a nuclear Hilbertian (CN) space. The space D is a nuclear Hilbertian (CUCN) space such that each component space is a nuclear countably Hilbert space. Further, in the case of space D, there is a sequence of non-degenerate positive hermitian forms ( , )_{m} (m = 0, 1, 2, …) defined on D such that in each component space X_{n} (n = 1, 2, …), we have (x, y)_{m}^{n} = (x, y)_{m} for x, y ∈ X_{n} and m = 0, 1, 2, … ([18, p. 332]).

(c) Nuclear Hilbertian (CUN) space such that each component space is a Hilbert space. Each component space X_{n} (n = 1, 2, …) is a Hilbert space endowed with a non-degenerate positive hermitian form ( , )_{n} for each n. In this case, for every n there exists an m > n such that the embedding of X_{n} into X_{m} is a nuclear mapping. The space S′ is such a typical space ([18, p. 333]).

(d) Nuclear Hilbertian (UCs-N) space such that Σ is uncountable and each component space is not a nuclear Hilbertian (Cs-N) space. The space D′ is such a typical space ([18, p. 334]).

III. Bochner, McShane and Henstock-Kurzweil Integrals and differentiations of (Cs-N) space valued functions

Consider a (Cs-N) space (X, {p_{m}}_{m=0}^{∞}) (a (UCs-N) space with a single component space). Then, {p_{m}}_{m=0}^{∞} is a sequence of seminorms such that p_{0}(x) ≤ p_{1}(x) ≤ ⋯ for x ∈ X. Put N(m) = {x ∈ X : p_{m}(x) = 0}. Then, the quotient space X/N(m) is a normed space. We denote the element of the quotient space with x ∈ X as a representative by [x]_{m} . We denote the completion of the normed space X/N(m) by (X̂_{m}, p̂_{m}), where p̂_{m} denotes the norm on X̂_{m}. In particular, we denote the element of X̂_{m} with a Cauchy sequence {[x]_{m}, [x]_{m}, …} (x ∈ X) as a representative by {[x]_{m}}ˆ . For an Xvalued function f, we define X̂_{m}-valued function f̂_{m} by f̂_{m}(t) = {[f(t)]_{m}}ˆ .

1. Bochner integral of (Cs-N) space valued function.

Let (X, {p_{m}}_{m=0}^{∞}) be a separated (Cs-N) space, and let (S, M, μ) be a non-negative finite valued measure space such that S is an abstract set, M is a σ-algebra of subsets of S, and μ is a real valued non-negative countably additive measure on M. An X-valued function f on S is called simple if there exist x(1), x(2), …, x(n) ∈ X and A(1), A(2), …, A(n) ∈ M such that f = ∑_{i=1}^{n} x(i)χ_{A(i)}, where χ_{A(i)}(x) = 1 if x ∈ A(i) and χ_{A(i)}(x) = 0 if x ∉ A(i). An X-valued function f on S is called μ-measurable if there exists a sequence of X-valued simple functions {f_{n}}_{n=1}^{∞} such that lim_{n→∞} f_{n}(t) = f(t) μ-almost everywhere on S.

Definition 4.1. Let X be a separated (Cs-N) space (X, {p_{m}}_{m=0}^{∞}). An X-valued function f on S is said to be Bochner integrable with respect to μ on S if f is μ-measurable and there exists a sequence of X-valued simple functions {f_{n}}_{n=1}^{∞} such that

(1) lim_{n→∞} ∫_{S} p_{m}(f_{n}(t) − f(t))dμ = 0 for m = 0, 1, 2, …,

(2) lim_{n→∞} ∫_{S} f_{n}(t)dμ exists in (X, {p_{m}}_{m=0}^{∞}). In this case, the integral of f on S, written ∫_{S} f(t)dμ, is defined by lim_{n→∞} ∫_{S} f_{n}(t)dμ.

In the definition above, if the space (X, {p_{m}}_{m=0}^{∞}) is complete, lim_{n→∞} ∫_{S} f_{n}(t)dμ necessarily exists. Hence, we obtain the following definition.

Definition 4.2. Let X be a separated and complete (Cs-N) space (X, {p_{m}}_{m=0}^{∞}). An X-valued function f on S is said to be Bochner integrable with respect to μ on S if f is μ-measurable and there exists a sequence of Xvalued simple functions {f_{n}}_{n=1}^{∞} such that lim_{n→∞} ∫_{S} p_{m}(f_{n}(t) − f(t))dμ = 0 for m = 0, 1, 2, . . .. In this case, the integral of f with respect to μ on S is defined by lim_{n→∞} ∫_{S} f_{n}(t)dμ.

Proposition 2. ([19, (0.20), p. 373]). Let (X, {p_{m}}_{m=0}^{∞}) be a separated complete (Cs-N) space. Let f be an X-valued function on S. Then, the function f is Bochner integrable with respect to μ on S as an (X, {p_{m}}_{m=0}^{∞})- valued function if and only if for m = 0, 1, 2, … the function f̂_{m} is Bochner integrable with respect to μ on S as an (X̂_{m}, p̂_{m})-valued function. In this case, {[∫_{a}^{b} f(t)dμ]_{m}}ˆ = ∫_{a}^{b} f̂_{m}(t)dμ for m = 0, 1, 2, . . . . .

2. McShane and Henstock-Kurzweil integrals of (Cs-N) space valued function.

Definition 5. Let X be a separated (Cs-N) space (X, {p_{m}}_{m=0}^{∞}), ν a non-decreasing real valued function on [a, b] and f an X-valued function on [a, b]. The function f is said to be McShane (resp. Henstock-Kurzweil ) integrable with respect to ν to z(∈ X) on [a, b] if for m = 0, 1, 2, … there exists a positive function δ_{m} on [a, b] such that for any δ_{m}-fine Lebesgue (resp. Perron) partition P of [a, b]: {([t_{0}, t_{1}], x_{1}), …, ([t_{p−1}, t_{p}], x_{p})}, we have

p_{m}(∑_{i=1}^{p} f(x_{i})(ν(t_{i}) − ν(t_{i−1})) − z) < 1/2^{m}.

Such a z is uniquely determined ([18, Lemma 4, p. 320]). In this case, the integral of f with respect to ν on [a,b], written ∫_{a}^{b} f(t)dν, is z. Simply, we set

σ(f, P; ν) = ∑_{i=1}^{p} f(x_{i})(ν(t_{i}) − ν(t_{i−1})).

Proposition 3. Let X be a separated complete (Cs-N) space (X, {p_{m}}_{m=0}^{∞}). Let f be an X-valued function on [a, b]. Then, the function f is McShane (resp. Henstock-Kurzweil ) integrable with respect to ν on [a, b] as an (X, {p_{m}}_{m=0}^{∞})-valued function if and only if for m = 0, 1, 2, …, the function f̂_{m} is McShane (resp. Henstock-Kurzweil ) integrable with respect to ν on [a, b] as an (X̂_{m}, p̂_{m})-valued function. In this case, {[∫_{a}^{b} f(t)dν]_{m}}ˆ = ∫_{a}^{b} f̂_{m}(t)dν for m = 0, 1, 2, ….

Proof. We prove only for the case of Henstock-Kurzweil integral. The “only if” part follows from [18, Lemma 6, p. 322]. The “if” part: Suppose that f̂_{m} is Henstock-Kurzweil integrable with respect to ν to an element of X̂_{m} , say ẑ_{m} , on [a, b] as an (X̂_{m}, p̂_{m})-valued function. Then, there exists a positive function δ_{m} on [a, b] such that for all δ_{m}-fine Perron partitions P of [a, b] we have

p̂_{m}(σ(f̂_{m}, P; ν) − ẑ_{m}) < 1/2^{m+1}.

Since the set {{[x]_{m}}ˆ : x ∈ X} is dense in (X̂_{m}, p̂_{m}), there exists an x_{m} ∈ X such that

p̂_{m}(ẑ_{m} − {[x_{m}]_{m}}ˆ ) < 1/2^{m+1}.

Therefore, for all δ_{m}-fine Perron partition P of [a, b] we have

p̂_{m}(σ(f̂_{m}, P; ν) − {[x_{m}]_{m}}ˆ ) < 1/2^{m}.

Now, fix a non-negative integer n and let m ≥ n. Then, we have

p̂_{n}(σ(f̂_{n}, P; ν) − {[x_{m}]_{n}}ˆ ) ≤ p̂_{m}(σ(f̂_{m}, P; ν) − {[x_{m}]_{m}}ˆ ) < 1/2^{m}

for all δ_{m}-fine Perron partitions P of [a, b]. Hence, by Cousin’s lemma, if k, m ≥ n,

p̂_{n}({[x_{k}]_{n}}ˆ − {[x_{m}]_{n}}ˆ ) < 1/2^{k} + 1/2^{m},

and therefore

p_{n}(x_{k} − x_{m}) < 1/2^{k} + 1/2^{m}.

Hence, the sequence {x_{m}}_{m=0}^{∞} in X is a Cauchy sequence under the seminorm p_{n} for n = 0, 1, 2, …. Therefore, by (C_{2}) the sequence is a Cauchy sequence in the space (X, {p_{m}}_{m=0}^{∞}). Since the space is complete, the limit lim_{m→∞} x_{m} = x exists in (X, {p_{m}}_{m=0}^{∞}). Now, we have

p_{n}(x_{n+1} − x) ≤ 1/2^{n+1}.

On the other hand, for all δ_{n+1}-fine Perron partitions P of [a, b], where δ_{n+1} is the one chosen in the above, we have

p_{n}(σ(f, P; ν) − x_{n+1}) < 1/2^{n+1}.

Consequently, for all δ_{n+1}-fine Perron partitions P of [a, b] we have

p_{n}(σ(f, P; ν) − x) < 1/2^{n}.

Thus, f is Henstock-Kurzweil integrable with respect to ν on [a, b] to x as an (X, {p_{m}}_{m=0}^{∞})-valued function.

3. Differentiation of (Cs-N) space valued function. Let X be a separated (Cs-N) space (X, {p_{m}}_{m=0}^{∞}) and f an X-valued function on [a, b]. Let t ∈ [a, b]. If there exists an element x ∈ X such that for m = 0, 1, 2, …, there exists δ_{m} > 0 such that p_{m}((f(s) − f(t))/(s − t) − x) < 1/2^{m} whenever 0 < |s − t| < δ_{m} (s ∈ [a, b]), then the function f is said to be differentiable at t . Since X is separated, such an element x is uniquely determined, and we denote the element x by f′(t).

Proposition 4. Let X be a separated complete (Cs-N) space (X, {p_{m}}_{m=0}^{∞}) and f an X-valued function defined on [a, b]. Then, the function f is differentiable at t ∈ [a, b] if and only if, for m = 0, 1, 2, …, f̂_{m} is differentiable at t as an (X̂_{m}, p̂_{m})-valued function. In this case, we have (f̂_{m})′(t) = {[f′(t)]_{m}}ˆ for m = 0, 1, 2, ….

Proof. The “only if” part is clear. The “if” part: For a given non-negative integer m, there exists x̂_{m} ∈ X̂_{m} such that we can find a δ(m) > 0 so that p̂_{m}((f̂_{m}(s) − f̂_{m}(t))/(s − t) − x̂_{m}) < 1/2^{m} whenever 0 < |s − t| < δ(m) (s ∈ [a, b]). Since the set {{[x]_{m}}ˆ : x ∈ X} is dense in (X̂_{m}, p̂_{m}), there exists an x_{m} ∈ X such that

p̂_{m}((f̂_{m}(s)−f̂_{m}(t))/(s−t)−{[x_{m}]_{m}}ˆ ) < 1/2^{m} if 0 < |s−t| < δ(m) (s ∈ [a, b]).

Hence, we have

p_{m}((f(s) − f(t))/(s − t) − x_{m}) < 1/2^{m} if 0 < |s − t| < δ(m) (s ∈ [a, b]).

Therefore, for any n ≥ m , we have

p_{m}((f(s) − f(t))/(s − t) − x_{n}) < 1/2^{n} if 0 < |s − t| < δ(n) (s ∈ [a, b]),

because p_{n}(x) ≥ p_{m}(x) in X. Fix a non-negative integer m. Given ε > 0, take an n(ε) with n(ε) > m and 1/2^{n(ε)} < ε/2. Then, for h and k with h, k ≥ n(ε) we have p_{m}(x_{h} − x_{k}) < ε . Hence, {x_{k}}_{k=0}^{∞} is a fundamental sequence in (X, p_{m}) for m = 0, 1, 2, …, and therefore in (X, {p_{m}}_{m=0}^{∞}). Since the space (X, {p_{m}}_{m=0}^{∞}) is complete, the limit lim x_{k} = x exists in (X, {p_{m}}_{m=0}^{∞}). Now, for m = 0, 1, 2, …, take an n = n(m) such that n > m and p_{m}(x_{n} − x) < 1/2^{m+1}, then

\begin{aligned}& p_{m}((f(s) − f(t))/(s − t) − x) \\ ≤ & p_{m}((f(s) − f(t))/(s − t) − x_{n}) + p_{m}(x_{n} − x) < 1/2^{m} \end{aligned}

if 0 < |s − t| < δ(n) (s ∈ [a, b]). Thus, f is differentiable at t as an (X, {p_{m}}_{m=0}^{∞})-valued function, and f′(t) = x. The last part of the proposition is clear from the demonstration above.

IV. (UCs-N) space valued Bochner, McShane and Henstock-Kurzweil integrals

1. (UCs-N) space valued Bochner integrals.

Definition 6. Let X be a separated (UCs-N) space with component spaces (X_{α}, {p_{m}^{α}}_{m=0}^{∞}) (α ∈ Σ) and let (S, M, μ) be a non-negative finite valued measure space. We say an X-valued function f on S is Bochner integrable with respect to μ to z on S if there exists a component space X_{α} such that:

(1) The image of S by f is contained in X_{α} and z ∈ X_{α}.

(2) The function f is Bochner integrable with respect to μ on S to z as an (X_{α}, {p_{m}^{α}}_{m=0}^{∞})-valued function.

When it is necessary to indicate explicitly such a space X_{α} , we say the function f is Bochner integrable(X_{α}) with respect to μ. Definition 6 is well defined by the facts that Σ is upward directed, and if f is Bochner integrable(X_{α}) with respect to μ to z on S, then f is Bochner integrable(X_{β}) with respect to μ to the same element z on S for every β with β ≥ α. In particular, we say an X-valued function f on S is μ-measurable(X_{α}) on S if the image of S by f is contained in X_{α} and f is μ-measurable on S as an (X_{α}, {p_{m}^{α}}_{m=0}^{∞})-valued function. (Cf. [26, Definition 5, p. 75] for example.)

When ([a, b], M, μ) is a measure space such that M is the family of Lebesgue measurable subsets of [a, b] and μ is Lebesgue measure on M, we say f is Bochner integrable with respect to Lebesgue measure, or simply, Bochner integrable, or measurable, etc. omitting “with respect to Lebesgue measure”.

Proposition 5. Let X be a separated (UCs-N) space. Then, any simple function is Bochner integrable with respect to μ on S.

Proposition 6. ([14, Proposition 21, p. 120], [18, p. 324]). If X is a complete separated (Cs-N) space, then any continuous X-valued function on [a, b] is Bochner integrable on [a, b].

Proposition 7. Let X be a separated (UCs-N) space with complete component spaces for which the statement (UCs-N,c) is true. Then, any continuous X-valued function on [a, b] is Bochner integrable on [a, b].

2. (UCs-N) space valued McShane and Henstock-Kurzweil integrals.

Definition 7. Let X be a separated (UCs-N) space with component spaces (X_{α}, {p_{m}^{α}}_{m=0}^{∞}) (α ∈ Σ) and ν a non-decreasing real valued function on [a, b]. An X- valued function f on [a, b] is said to be McShane (resp. Henstock-Kurzweil) integrable with respect to ν to z on [a, b] if there exists a component space X_{α} such that:

(1) The image of [a, b] by f is contained in X_{α} and z ∈ X_{α}.

(2) The function f is McShane (resp. Henstock-Kurzweil) integrable with respect to ν to z on [a, b] as an (X_{α}, {p_{m}^{α}}_{m=0}^{∞})-valued function.

When it is necessary to indicate explicitly such an X_{α}, we say f is Mc-Shane (resp. Henstock-Kurzweil) integrable(X_{α}) with respect to ν. Definition 7 is well defined by the same reason as in Definition 6. In particular, when ν is the identity function on the real space R, i.e., ν(t) = t on R, we simply say f is McShane (resp. Henstock-Kurzweil) integrable, omitting “with respect to ν”.

Proposition 8. ([22, Proposition 13, p. 268]). Let X be a separated (UCs-N) space. If an X-valued function f is Bochner integrable on [a, b], then it is McShane integrable on [a, b], and both integrals coincide.

Proposition 9. Let X be a separated (UCs-N) space. A simple Xvalued function on [a, b] is Bochner and McShane integrable on [a, b] and both integrals coincide.

Proposition 10. Let X be a complete separated (Cs-N) space. Any continuous X-valued function on [a, b] is Bochner and McShane integrable on [a, b] and both integrals coincide.

Proposition 11. Let X be a separated (UCs-N) space with complete component spaces for which the statement (UCs-N,c) is true. Then, any continuous X-valued function on [a, b] is Bochner and McShane integrable on [a, b] and both integrals coincide.

Example 5

By Example 2, S, S′, D and D′-valued continuous functions are Bochner and McShane integrable on [a, b] and both integrals coincide.

V. McShane and Henstock-Kurzweil integrals for functions taking values in a nuclear Hilbertian (UCs-N) space

We first show the following two key lemmas which are pivotal in our theory.

Key Lemma 1. Let X be a Hilbert space with a scalar product ( , ), and p the norm associated with ( , ). Let z_{i} ∈ X and let a_{i} be real numbers for i = 1, 2, …, q, and x ∈ X. Then

| ∑_{i=1}^{q}(a_{i}z_{i}, x)| ≤ p(∑_{i=1}^{q} a_{i}z_{i})p(x).

For the converse of Key Lemma 1, we have the following idea.

Key Lemma 2. Let X_{α} and X_{β} be Hilbert spaces with scalar products ( , )_{α} and ( , )_{β}, and p_{α} and p_{β} the norms associated with ( , )_{α} and ( , )_{β}, respectively. Let T_{β}^{α} be a nuclear mapping of the space X_{α} into X_{β} , i.e., a mapping having the form:

N(α, β) : T_{β}^{α}(x) = ∑_{k=1}^{∞} λ_{βk}^{α}(x, e_{k}^{α})_{α}e_{k}^{β} for x ∈ X_{α},

where {e_{k}^{α}}_{k=1}^{∞} and {e_{k}^{β}}_{k=1}^{∞} are orthonormal systems of vectors in X_{α} and X_{β}, respectively, λ_{βk}^{α} > 0, and ∑_{k=1}^{∞} λ_{βk}^{α} < ∞. Then, the following inequality holds for any finite sequence z_{i} ∈ X_{α} (i = 1, 2, …, p).

∑_{i=1}^{p} p_{β}(T_{β}^{α}(z_{i})) ≤ ∑_{k=1}^{∞} λ_{βk}^{α}{∑_{i=1}^{p} |(z_{i}, e_{k}^{α})_{α}|}.

For example, we sketch here the proof of the following lemma, because it is a typical way of proofs of propositions in the nuclear space valued case.

Lemma 1. Under the same assumption as in Key Lemma 2, the following holds. Let f be an X_{α} -valued McShane (resp. Henstock-Kurzweil) integrable function on [a, b], and let F be an indefinite integral of f. If δ is a positive function on [a, b] such that

p_{α}(∑_{i=1}^{p} f(x_{i})(t_{i} − t_{i−1}) − F ([t_{i−1}, t_{i}])) < ε/(3 ∑_{k=1}^{∞} λ_{βk}^{α})

for any δ-fine Lebesgue (resp. Perron) partition

{([t_{0}, t_{1}], x_{1}), …, ([t_{p−1}, t_{p}], x_{p})}

of [a, b], then for the same δ we have

∑_{i=1}^{p} p_{β}(T_{β}^{α}(f(x_{i}))(t_{i} − t_{i−1}) − T_{β}^{α}(F ([t_{i−1}, t_{i}]))) < ε

for any δ-fine Lebesgue (resp. Perron) partition

{([t_{0}, t_{1}], x_{1}), …, ([t_{p−1}, t_{p}], x_{p})}

of [a, b].

Proof. First we use the idea of Key Lemma 1. Put

z_{i} = f(x_{i})(t_{i} − t_{i−1}) − F ([t_{i−1}, t_{i}]) for i = 1, 2, …, p.

For k = 1, 2, …, we have

| ∑_{i=1}^{p}((f(x_{i}), e_{k}^{α})_{α}(t_{i} − t_{i−1}) − (F ([t_{i−1}, t_{i}]), e_{k}^{α})_{α}|
< p_{α}(∑_{i=1}^{p} f(x_{i})(t_{i} − t_{i−1}) − F ([t_{i−1}, t_{i}])) (by Key Lemma 1 )
< ε/(3 ∑_{k=1}^{∞} λ_{βk}^{α}).

Using the property in the real valued case (which appears in the proof of Saks-Henstock lemma in the real valued case), we have

∑_{i=1}^{p} |(f(x_{i}), e_{k}^{α})_{α}(t_{i} − t_{i−1}) − (F ([t_{i−1}, t_{i}]), e_{k}^{α})_{α}| < ε/(∑_{k=1}^{∞} λ_{βk}^{α}).

Hence, using Key Lemma 2, we have

∑_{i=1}^{p} p_{β}(T_{β}^{α}(f(x_{i}))(t_{i} − t_{i−1}) − T_{β}^{α}(F ([t_{i−1}, t_{i}])))

≤ ∑_{k=1}^{∞} λ_{βk}^{α}{∑_{i=1}^{p} |(f(x_{i}), e_{k}^{α})_{α}(t_{i} − t_{i−1}) − (F ([t_{i−1}, t_{i}]), e_{k}^{α})_{α}|}
< ε.

VI. Nuclear Hilbertian (UCs-N) space valued McShane and Henstock-Kurzweil integrals

In what follows, unless mentioned otherwise X denotes a nuclear Hilbertian (UCs-N) space with component spaces (X_{α}, {p_{m}^{α}}_{m=0}^{∞}) (α ∈ Σ) which is defined in Definition 3. The following propositions are proved. The idea of the proof is analogous to that of the corresponding proposition in Section 3 of [19] except Proposition 25. The idea of the proof of Proposition 25 is analogous to that of Proposition 12 of [21].

Proposition 13 (Saks-Henstock lemma). Let f be an X-valued Mc-Shane (resp. Henstock-Kurzweil) integrable function on [a, b], and F an indefinite integral of f. Then, there exists an α ∈ Σ such that the function f is McShane (resp. Henstock-Kurzweil) integrable(X_{α}) and for m = 0, 1, 2, …, there exists a positive function δ_{m} on [a, b] such that for any δ_{m}-fine Lebesgue (resp. Perron) partition {([c_{1}, d_{1}], x_{1}), …, ([c_{p}, d_{p}], x_{p})} in [a, b] we have

∑_{i=1}^{p} p_{m}^{α}(f(x_{i})(d_{i} − c_{i}) − F ([c_{i}, d_{i}])) < 1/2^{m}.

Proposition 14. If an X-valued function f is Henstock-Kurzweil integrable on [a, b], then there exists an α ∈ Σ such that f is measurable(X_{α}) on [a, b].

Proposition 15. An X-valued function f is McShane integrable on [a, b] if and only if it is Bochner integrable on [a, b], and both integrals coincide.

Proposition 16. Let f be an X-valued Henstock-Kurzweil integrable function on [a, b]. Then, there exists an α ∈ Σ such that if β ≥ α, for m = 0, 1, 2, …, p_{m}^{β}(f(t)) is measurable on [a, b] and

p_{m}^{β}(∫_{c}^{d} f(t)dt) ≤ ∫_{c}^{d} p_{m}^{β}(f(t))dt for any [c, d] ⊂ [a, b].

A division of an interval [a, b] is a finite collection of non-overlapping intervals whose union is [a, b]. Let X be a semi-normed space (X, p) with a semi-norm p, and let F be an X-valued function. The variation of F on [a, b] is the extended real number

V (F, [a, b]) = sup(∑_{i=1}^{p} p(F ([t_{i−1}, t_{i}]))),

where the supremum is taken over all divisions {[t_{0}, t_{1}], [t_{1}, t_{2}], …, [t_{p−1}, t_{p}]} of [a, b]. If V (F, [a, b]) < ∞, we say that F is of bounded variation on [a, b]. When the space X is a (Cs-N) space (X, {p_{m}}_{m=0}^{∞}), we say that an X-valued function F is of bounded variation on [a, b] if F is of bounded variation on [a, b] as an (X, p_{m})-valued function for m = 0, 1, 2, . . .. When X is a (UCs-N) space with component spaces (X_{α}, {p_{m}^{α}}_{m=0}^{∞}) (α ∈ Σ), we say that an X-valued function F is of bounded variation on [a, b] if there exists an α such that the image of [a, b] by F is contained in X_{α} and F is of bounded variation on [a, b] as an (X_{α}, {p_{m}^{α}}_{m=0}^{∞})-valued function.

Proposition 17. Let f be an X-valued Henstock-Kurzweil integrable function on [a, b] and F an indefinite integral of f. Then, f is McShane integrable on [a, b] if and only if F is of bounded variation on [a, b]. In this case, there exists an α such that for m = 0, 1, 2, …, p_{m}^{α}(f(t)) is McShane integrable on [a, b] and ∫_{a}^{b} p_{m}^{α}(f(t))dt coincides with the variation on [a, b] of F treated as an (X_{α}, p_{m}^{α})-valued function.

1. Fundamental theorem of calculus

Let X be a separated (UCs-N) space with component spaces (X_{α}, {p_{m}^{α}}_{m=0}^{∞}) (α ∈ Σ) and let f be an X-valued function defined on [a, b]. Let t ∈ [a, b]. If there are an α and a neighborhood V (t) of t in [a, b] such that the image of V (t) by f is contained in X_{α} and f is differentiable at t as an (X_{α}, {p_{m}^{α}}_{m=0}^{∞})- valued function, then f is said to be differentiable at t, or differentiable(X_{α}) at t if necessary(see Chapter III, 3).

Proposition 18. Let f be an X-valued Henstock-Kurzweil integrable function on [a, b] and let F be an indefinite integral of f. Then, there exists an α such that F is differentiable(X_{α}) almost everywhere on [a, b] and F′(t) = f(t).

Proposition 19. Let F be an X-valued function on [a, b]. Suppose that there exists an α such that the image of [a, b] by F is contained in X_{α}, and F is continuous on [a, b] as an (X_{α}, {p_{m}^{α}}_{m=0}^{∞})-valued function and differentiable(X_{α}) except possibly at a countable set D ⊂ [a, b]. Then, the function F′(t) defined to be t he derivative of F at t if t ∈ [a, b] − D and an arbitrary element of X_{α} if t ∈ D is Henstock-Kurzweil integrable on [a, b] and ∫_{a}^{b} F′(t)dt = F (b) − F (a).

2. Descriptive definition

Let X be a (Cs-N) space (X, {p_{m}}_{m=0}^{∞}). An X-valued function F on [a, b] is called AC_{∗} if, given a negligible set E in [a, b] (i.e. a set E of measure zero) and a non-negative integer m, there exists a positive function δ_{m} on E such that ∑_{i=1}^{p} p_{m}(F ([c_{i}, d_{i}]) < 1/2^{m} for any Perron partition {([c_{1}, d_{1}], x_{1}), …, ([c_{p}, d_{p}], x_{p})} in [a, b] (anchored in E) that is δ_{m}-fine. When X is a (UCs-N) space with component spaces (X_{α}, {p_{m}^{α}}_{m=0}^{∞}) (α ∈ Σ), an Xvalued function F is called AC_{∗}, or AC_{∗}(X_{α}) if necessary, if there exists an α such that the image of [a, b] by F is contained in X_{α} and F is AC_{∗} on [a, b] as an (X_{α}, {p_{m}^{α}}_{m=0}^{∞})-valued function.

Proposition 20. Let F be an X-valued function on [a, b]. Suppose that there exists an α such that the image of [a, b] by F is contained in X_{α} and F is differentiable(X_{α}) almost everywhere on [a, b]. Let F′ be an X_{α}-valued function on [a, b] defined to be the derivative of F except for a set A of measure zero. Then, the following holds.

(1) If F′ is Henstock-Kurzweil integrable on [a, b] and F is an indefinite integral of F′, then the function F is AC_{∗} on [a, b].

(2) If F is AC_{∗} on [a, b], then F′ is Henstock-Kurzweil integrable on [a, b] and F is an indefinite integral of F′.

Proposition 21 (Descriptive definition). Let f be an X-valued function on [a, b]. The function f is Henstock-Kurzweil integrable on [a, b] if and only if there exists an X-valued function F on [a, b] such that

(1) there exists an α such that F is differentiable(X_{α}) almost everywhere on [a, b] and F′(t) = f(t),

(2) F is AC_{∗} on [a, b]. In this case, F is an indefinite integral of f.

This follows from Propositions 18 and 20.

3. Convergence theorem

Proposition 22. Let f be an X-valued function on [a, b] for which there exists an α such that f is Henstock-Kurzweil integrable(X_{α}) on [a, c] for every c ∈ (a, b). Suppose that lim_{c→b−} ∫_{a}^{c} fdt exists in (X_{β}, {p_{m}^{β}}_{m=0}^{∞}) for some β and the limit is z. Then, f is Henstock-Kurzweil integrable to z on [a, b].

Proposition 23. Let {f_{j}}_{j=1}^{∞} be a sequence of X-valued functions such that for some α each f_{j} is McShane (resp. Henstock-Kurzweil) integrable(X_{α}) on [a, b]. Suppose that there exists a β such that:

(1) The image of [a, b] by f_{j} is contained in X_{β} for every j, and there exists an X_{β}-valued function f such that lim_{j→∞} f_{j}(t) = f(t) as (X_{β}, {p_{m}^{β}}_{m=0}^{∞})- valued functions almost everywhere on [a, b].

(2) ∫_{a}^{b} p_{m}^{β}(f_{j}(t) − f_{k}(t))dt → 0 as j, k → ∞ for m = 0, 1, 2, …. Then, for some γ, f is McShane (resp.Henstock-Kurzweil) integrable(X_{γ}) on [a, b] and for any figure B ⊂ [a, b],

lim_{j→∞} ∫_{B} f_{j}(t)dt = ∫_{B} f(t)dt in (X_{γ}, {p_{m}^{γ}}_{m=0}^{∞}),

where the convergence is uniform in B.

Proposition 24. Let {f_{j}}_{j=1}^{∞} be a sequence of X-valued functions on [a, b]. Suppose that:

(1) For some α, f_{j} is McShane (resp. Henstock-Kurzweil) integrable(X_{α}) on [a, b] for j = 1, 2, ….

(2) There exists a β′ such that for any β ≥ β′ and any e ∈ X_{β} with p_{m}^{β}(e) = 1, the following (∗) holds for m = 0, 1, 2, ….

(∗) (f_{1}(t), e)_{m}^{β} ≤ (f_{2}(t), e)_{m}^{β} ≤ ⋯ almost everywhere on [a, b].

(3) The limit lim_{j→∞} ∫_{a}^{b} f_{j}(t)dt exists in X.

(4) For some γ, the limit lim_{j→∞} f_{j}(t) = f(t) exists in (X_{γ}, {p_{m}^{γ}}_{m=0}^{∞}) for almost all t ∈ [a, b]. Then, there exists a κ such that f is McShane (resp. Henstock-Kurzweil) integrable (X_{κ}) on [a, b] and for any figure B in [a, b]

lim_{j→∞} ∫_{B} f_{j}(t)dt = ∫_{B} f(t)dt in (X_{κ}, {p_{m}^{κ}}_{m=0}^{∞}),

where the convergence is uniform in B.

Proposition 25 (Convergence theorem [4, Theorem 9.1]). Let f_{j} (j = 1, 2, …) and f be X-valued functions on [a, b]. Suppose that for some α:

(1) f_{j} is Henstock-Kurzweil integrable(X_{α}) on [a, b] for j = 1, 2, ….

(2) lim_{j→∞} f_{j}(t) = f(t) almost everywhere on [a, b] as (X_{α}, {p_{m}^{α}}_{m=0}^{∞})- valued functions.

(3) For m = 0, 1, 2, … there are an M_{m} ≥ 0 and a positive function δ_{m} on [a, b] such that

p_{m}^{α}(∑_{i=1}^{p} f_{j(x(i))}(x(i))(t_{i} − t_{i−1})) ≤ M_{m}

for all δ_{m}-fine Perron partitions {([t_{0}, t_{1}], x(1)), …, ([t_{p−1}, t_{p}], x(p))} of [a, b] and all choices of positive integer valued functions j(t) on [a, b]. Then for some β with β ≥ α, f is Henstock-Kurzweil integrable(X_{β}) on [a, b] and for any figure B ⊂ [a, b]

lim_{j→∞} ∫_{B} f_{j}dt = ∫_{B} fdt as (X_{β}, {p_{m}^{β}}_{m=0}^{∞})-valued functions,

where the convergence is uniform in B.

4. Integration by parts

Proposition 26. Let f be a Henstock-Kurzweil integrable X-valued function on [a, b] and G a non-decreasing real valued function on [a, b]. Let F (t) = ∫_{a}^{t} fdt (t ∈ [a, b]). Then, fG is Henstock-Kurzweil integrable on [a, b], F is McShane integrable with respect to G on [a, b] and

∫_{a}^{b} fGdt = F (b)G(b) − ∫_{a}^{b} F dG.

Proposition 27. Let f be a McShane (resp. Henstock-Kurzweil) integrable X-valued function on [a, b] and g a McShane (resp. Henstock-Kurzweil ) integrable real valued function on [a, b]. Let F (t) = ∫_{a}^{t} fdt and G(t) = ∫_{a}^{t} gdt. Then, fG + F g is McShane (resp. Henstock-Kurzweil) integrable on [a, b], and

∫_{a}^{b}(fG + F g)dt = F (b)G(b).

5. Ordinary differential equations

Let X be a vector space,

g : [a, b] × E → X, where E ⊂ X,
x′ = g(t, x).

The methods of the Carathéodory and Henstock solutions for the ordinary differential equations of the type above shown in the case when X is the real space can be extended to the case when X is a nuclear countably Hilbert space, i.e., nuclear Hilbertian (CN) space. See [20] and [21] in detail.

Finally we propose the following problem.

Problem. Develop the theory analogous to our theory to the case when the space X is treated as a topological vector space.

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[13] S. Nakanishi: On topological structures of spaces C([0, 1], X) and C([0, ∞), X), I, II, III, Math. Japon., 30,31,32 (1985, 1986,1987), 485-493, 75-84, 957-970.

[14] S. Nakanishi: Integration of ranked vector space valued functions, Math. Japon., 33 (1988), 105-128.

[15] S. Nakanishi: Ranked vector space valued measures and the Radon-Nikodym theorem, Math. Japon., 34 (1989), 253-274.

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[17] S. Nakanishi: The Henstock integral for functions with values in nuclear spaces and the Henstock lemma, Journal of Math. Study, 27 (1994), 133-141.

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[22] K. Sakurada and S. Nakanishi: Equivalence of the McShane and Bochner integrals for functions with values in Hilbertian (UCs-N) spaces endowed with nuclearity, Math. Japon., 47 (1998), 261-272.

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Professor Emeritus of Osaka Prefecture University

4-15 Jonanteramachi, Tennoji-Ku, Osaka 543-0017, Japan