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The concept of a function of bounded ‐variation generalizes the concepts of a function of semivariation and of a function of bounded variation, as we will see in the sequel.

Definition 1.20: A bilinear triple (we write BT) is a set of three vector spaces , , and , where and are normed spaces with a bilinear mapping . For and , we write , and we denote the BT by or simply by . A topological BT is a BT , where is also a normed space and is continuous.

If and are normed spaces, then we denote by the space of all linear continuous functions from to . We write and , where denotes the real line. Next, we present examples, borrowed from [127], of bilinear triples.

Example 1.21: Let , , and denote Banach spaces. The following are BT:

1 , , , and ;

2 , , , and ;

3 , , , and ;

4 , and .

Given a BT , we define, for every , a norm

and we set . Whenever the space is endowed with the norm , we say that the topological BT is associated with the BT .

Let be a vector space and be a set of seminorms defined on such that implies Then, defines a topology on , and the sets form a basis of neighborhoods of 0. The sets form a basis of the neighborhood of . Moreover, when endowed with this topology, is called a locally convex space (see [127], p. 3, 4).

Example 1.22: Every normed or seminormed space is a locally convex space.

For other examples of locally convex spaces, we refer to [110].

Definition 1.23: Given a BT , and a function , for every division , we define

Then, is the ‐variation of on . We say that is a function of bounded ‐variation, whenever . In this case, we write .

The following properties are not difficult to prove. See, e.g. [127, 4.1 and 4.2].

1 (SB1) is a vector space and the mapping is a seminorm.

2 (SB2) Given , the function is increasing.

3 (SB3) Given and , .

Definition 1.24: Consider the BT . Then, instead of and , we write simply and , respectively. Hence,

and we call any element of a function of bounded semivariation.

Definition 1.25: Given a function , a normed space, and a division , we define

and the variation of is given by

If , then is called a function of bounded variation, in which case, we write .

It is not difficult to prove that and .

Moreover (see [127, Corollary I.3.4]), .

The space is complete when equipped with the variation norm, , given by

for When there is no room for misunderstanding, we may use the notation instead of .

Remark 1.26: Consider a BT . The definition of variation of a function , where is a normed space, can also be considered as a particular case of the ‐variation of in two different ways.

1 Let , or and . By the definition of the norm in , we haveThus, when we consider the BT , we write and instead of and , respectively.

2 Let , or and . By the Hahn–Banach Theorem, we haveand, hence,

Definition 1.27: Given , we define the spaces

Such spaces are complete when endowed, respectively, with the norm given by the variation and the norm given by the semivariation

where

and is a division of .

The following properties are not difficult to prove:

1 (V1) Every is bounded and , .

2 (V2) Given and , we have .

Remark 1.28: Note that property (V1) above implies for all .

For more details about the spaces in Definition 1.27, the reader may want to consult [127]. The next results are borrowed from [126]. We include the proofs here since this reference is not easily available. Lemmas 1.29 and 1.30 below are, respectively, Theorems I.2.7 and I.2.8 from [126].

Lemma 1.29: Let . Then,

1 For all , there exists .

2 For all , there exists .

Proof. We only prove item (i), because item (ii) follows analogously. Consider an increasing sequence in converging to . Then,

for all Therefore, we have and, hence, as Thus, is a Cauchy sequence, since for any given , we have

for sufficiently large . Finally, note that the limit of is independent of the choice of , and we finish the proof.

It comes from Lemma 1.29 that all functions of bounded variation are also regulated functions (see, e.g. [127, Corollary 3.4]) which, in turn, are Darboux integrable [127, Theorem 3.6].

Lemma 1.30: Let . For every , let . Then,

1 , ;

2 , .

Proof. By property (SB2), is increasing and, hence, and exist. By Lemma 1.29, and also exist. We prove (i). The proof of (ii) follows analogously.

Suppose . Then, property (V2) implies Thus,

and, hence,

Conversely, for any given , let . Then for every , there exists such that and , and there exists a division such that

Then,

and, hence, which completes the proof.

Using the fact that , the following corollary follows immediately from Lemma 1.30.

Corollary 1.31: Let . Then the sets

are finite for every .

Thus, we have the next result which can be found in [126, Proposition I.2.10].

Proposition 1.32: Let . Then the set of points of discontinuity of is countable.

Let us define

A proof that equipped with the variation norm, , is complete can be found in [126, Theorem I.2.11]. We reproduce it in the next theorem.

Theorem 1.33: , equipped with the variation norm, is a Banach space.

Proof. We know that , with the variation norm, is a Banach space. Let be a sequence in converging to in the variation norm. Then, since for every and every , , we obtain

which tends to zero as . Hence, .

We end this section with the Helly's choice principle for Banach space‐valued functions due to C. S. Hönig. See [127, Theorem I.5.8].

Theorem 1.34 (Theorem of Helly): Let be a BT and consider a sequence of elements of , with , for all , and such that there exists , with for all and all . Then, and . Moreover, if , with , for all , then and .

Proof. Consider a division and let , with , for . Then, for all , we have

where the first member on the right‐hand side of the inequality is smaller than , since . Moreover, by hypothesis, given , there is such that, for all , , for Hence, for all ,

and we conclude the proof of the first part. The second part follows analogously.

For more details about functions of bounded variation, the reader may want to consult [68], for instance.