^{1, 2}

^{1}

^{2}

This paper studies typical Banach and complete seminormed
spaces of locally summable functions and their continuous functionals. Such
spaces were introduced long ago as a natural environment to study almost periodic
functions (Besicovitch, 1932; Bohr and Fölner, 1944) and are defined by boundedness of suitable

Families of Banach spaces of locally

All the spaces of bounded

For this goal, we focus our attention onto three significant families of locally

We give an expanded and revised presentation of some known results, mostly taken from the fundamental paper [

The Besicovitch-Marcinkiewicz spaces

Here is their definition. Let

Marcinkiewicz spaces have been studied or used in [

It is obvious that

However, let us show that the vector valued integral in (

Let

Therefore there exists a unique function

The function

Inequality (

Since

A related family of spaces are the bounded

Since

But first let us clarify the reason for which the length of the interval

The space of all functions

Indeed, it is obvious that if

As for the spaces introduced before, also

The translation operator

Without loss of generality, let

Stepanoff functions, introduced in [

More generally, for every

The norms

For every

On the other hand, let

If

It follows from Proposition

The Weyl norm defines a normed space called the Weyl space

It is easily seen that the spaces

The Marcinkiewicz spaces

Let

Let us choose a sequence

We claim that the sequence

For the goal of understanding duality, it is appropriate to discuss first the inclusions between all these spaces, and their structure.

First of all, it is obvious from the inclusions between

Let us come to more interesting inclusions. It is easy to see that

For every locally

If

In computing the

The first inequality is obvious. For the second, by splitting the interval

Recall that

The embedding of

As a consequence of Corollary

We have already observed that the latter quotient is embedded in the former, and, for every

Let

From every sequence

When we consider spaces of functions over disjoint intervals, for instance if

It follows from Lemma

We now consider another Banach space of functions with appropriate

We also recall that the null space

Now the following result, proved in [

(i) One has

(ii) Also

(iii) If

(iv) Moreover

(v) If

We give a sketch of the proof. Part (i) follows directly from the definition of

For

It is easy to see, as in [

It follows from the tensor product structure of

For all

We start by building a sequence of intervals with larger and larger distance and length. Start with

Let now

The Riesz representation theorem shows that all continuous linear functionals on

Let us consider the dual space of the Marcinkiewicz space

Indeed, all continuous linear functionals on a semi-normed complete space

Since every compactly supported

There are interesting instances of

Here are some other interesting

Since

Hölder’s inequality shows that, for

In the next sections we expand these ideas to achieve a more complete representation, developed in [

A normed (or semi-normed) space is uniformly convex if, for every

The following results are stated without proof in [

(i) Let

(ii) The same statement holds if

(iii) If, more generally,

We can restrict attention to the bidimensional subspace of

Part (i) follows by considering the segment

For part (ii), it is enough to observe that, whenever

To prove (iii) consider the triangle whose vertices are

Estimate for

For

Now we describe the dual of the spaces of bounded

We start with some easy comments on functionals that attain their norm.

(i) Let

(ii) Every real finitely additive finite Borel measure on a Borel space

(iii) If

(iv) Not every (countably additive) finite (real or complex) Borel measure on

We first observe that, for every

Then, for every