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The abstract of the talk (see below) may be downloaded in one of the following formats:
{\small
An asymmetric norm is a nonnegative and subadditive positively
homogeneous real function $q$ defined on a linear space $X$ satisfying that
$q(x)=q(-x)=0$ implies $x=0$.
An asymmetric normed linear space is a pair $(X,q)$ such that
$X$ is a linear space and $q$ is an asymmetric norm on $X$.
Such a function $q$ induces a quasi-metric $d_q$ by mean of the formula:
\[
d_q(x,y)=q(y-x), \quad x,y \in X.
\]
Thus, the study of the structure of asymmetric normed linear spaces
can be done from the topological point of view as a part of the
so called asymmetric topology. Such kind of spaces are
interesting because of their applications to the
Computational Complexity Theory.
In our lecture we present other point of view for the study of the
asymmetric normed linear spaces. Our framework is the Banach space
theory. Although an asymmetric norm does not define in general a
normed space, it is possible to apply certain techniques of the
Banach space theory to several problems. For instance,
many examples of asymmetric normed linear spaces are constructed
using lattice norms on sequence spaces or function spaces.
Thus, the theory of the Banach spaces
provides good tools to study the following problems:
\begin{enumerate}
\item
\emph{When an asymmetric linear space is Hausdorff.}
Following classical techniques of the Banach space theory we can give a
characterization of those asymmetric normed linear spaces that satisfies the
$T_2$ separation axiom. Moreover, a reasonable generalization of the concept of
complemented subspaces of Banach spaces leads to a canonical decomposition
of an asymmetric normed linear space as a direct sum of a Hausdorff space
and what we call a purely non Hausdorf asymmetric normed linear space.
\item
\emph{The definition of the dual of an asymmetric normed linear space
and the weak topologies.}
A great part of the properties of the normed
and Banach spaces are related to the weak topologies that can be defined
on them in a natural way. The set of continuous linear functions
on normed spaces gives a Banach space with its natural norm.
Although in the case of asymmetric normed linear spaces this set does not
define in general a linear space, it defines a semilinear space, and we can
define in a natural way the dual of an asymmetric normed linear space.
Thus, several (weak) topologies can be induced by the relation between
a space and its dual. For instance, we can prove an asymmetric version of
the Alaoglu Theorem on the compactness of the unit ball of the dual space
with respect to the corresponding weak* topology. This is one of the main
tools on Banach space theory. It is also possible to give, for a restricted
class of asymmetric normed linear spaces, a version of the
Hahn Banach Theorem.
\item
\emph{Extensions of asymmetric norms defined on semilinear spaces to
their linear covers.}
Some of the most interesting applications of the
theory of asymmetric normed linear spaces actually depends on
the properties of semilinear spaces that are subsets of these spaces
(for instance, the dual spaces).
Therefore, the natural question that arises what are the conditions that are
required to assure that there exists an extension of an asymmetric norm
from a semilinear subspace to its linear cover.
\end{enumerate}
}