Short about thick and thin sets

    Complete metric spaces are always very fat. Baire showed that complete metric spaces never are unions of sets which are so small that their closure contains no open set. Thus, if a complete metric space has been discovered to be a countable union of sets, we can be sure that the closure of one of the sets in the union contains a ball.

    A subset of a Banach space is called norming for the dual if its closed symmetric convex hull contains a ball centered at the origin. Such sets can occure in two massiveness variants, thick or thin:
 

 
Definition:      The norming set is called thin if it can be written as a countable, increasing union of non-norming sets, otherwise it is called thick.

 
Take as an example the unit vectors in the unit ball in l1. The closed symmetric convex hull of this set is the whole unit ball of l1. This set is countable, thus thin.
Take as another example any set of second category in a Banach space. Suppose it has been written as a countable, increasing union. Then, since it is of second category in the space, the closure of some set in the union contains a ball. Thus the closed symmetric convex hull of that set contains a ball centered at the origin. So any set of the second category is thick.


What's the point?

    Recall the uniform boundedness theorem and the open mapping theorem. From these theorems we can conclude that families of operators, pointwise bounded on sets of second category are uniformly bounded and that operators onto sets of second category are onto. A natural, and very central question is: Are there weaker conditions to build the uniform boundedness theorem and the open mapping theorem on than second category? Let us agree to say that such sets have the boundedness property and the surjectivity property, respectively.

    In 1930 Oleg Nikodym showed that the set of characteristic functions has the boundedness property (for a big class of functionals) in the space of bounded functions on [0,1]. This set is not of second category, neither is it's symmetric convex hull. The result of Nikodym was generalized to the space of bounded functions on any finite measure space with a sigma-algebra B.by Grothendieck in the fifties. In this extension the characteristic functions are proved to have the boundedness property for all functionals. In 1968 Seever discovered that in this space, the set of characteristic functions also has the surjectivity property.

    And no I am sure you have guessed my main result.
 

 
Theorem:  For a Banach space X the following are equivalent statements:
  1. The set A is thick.
  2. Any linear operator from a Banach space Y onto A, is onto X.
  3. Any family of linear operators which are pointwise bounded on A, is uniformly bounded.

    The equivalence of 1 and 2 was discovered by Vladimir Fonf and Mihail Kadets in 1982. The equivalence between this and 3, and thus the relation to the fundaments of Banach spaces, was discovered by me in 1996. Since then I have been looking for candidates to be interresting thick sets, i.e. sets which are not also of the second category.

    References for further reading can be found in my list of publications.

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