We prove that the cardinality of the space

Let

The corresponding normed space is built using the quotient space determined by the relation

It is a known fact that the space

Höning [

Afterwards, using the same technique used by Höning, Merino improves Höning’s results in the sense that those results arise as corollaries of the following theorem.

There exists no ultrabornological, infra-(

On the basis of Corollary

For the reader’s convenience, we restate some concepts about topological vector spaces that will be used later in this paper which can be found particularly in [

Let

It is said that

It is a known fact that if

A topological vector space

Let

The above concepts have other equivalent characterizations. However, the characterizations that appear in the preceding definition are the best for the goals of this paper.

Let

The definitions of

Although webbed spaces are a subclass of infra-(

Some stability properties that enjoy the convex-Suslin spaces are the following: the countable product of convex-Suslin spaces is convex-Suslin and every closed subspace of a convex-Suslin space is a convex-Suslin space. Moreover, these spaces play a version of Closed Graph Theorem as set forth below.

Let

Now, we establish the notations that we will use throughout this paper.

Let

All vector spaces that appear throughout this paper are considered over the field of the real or complex numbers.

The cardinality of

The cardinality of a Hamel base or algebraic base is denoted by

The symbols

We will denote the set of continuous functions with real values whose domain is the compact interval

The following result establishes necessary and sufficient conditions to determine when a vector space has a norm under which it is a Banach space. This result is the foundation of the goals of this paper.

Let

The cardinality of the space

Since the application

On the other hand, as the constant functions are Henstock-Kurzweil integrables, it follows that

There exists a norm on

It is a known fact that

Hereafter, the Alexiewicz topology and the topology induced by the norm of Proposition

The topology

Let

Although

A vector topology

Now, we shall prove that the space

Let

Let

Therefore

The space

On the basis of Proposition

On the basis of Corollary

On the other hand, employing the same technique that we have been using entails the following result.

The space

Suppose that

Since K-Suslin spaces are a subclass from convex-Suslin spaces, [

Although apparently the topology

There are other types of spaces which enjoy certain versions of Closed Graph Theorem, for example, the spaces: Suslin, quasi-Suslin, [

As another application of the technique which has been used in this paper and reiterating the known fact that

This paper is supported by the Committee of