  Non-Archimedean mathematics (in particular, nonstandard analysis) allows to
construct some useful models to study certain phenomena arising in PDE's; for
example, it allows to construct generalized solutions of differential equations
and variational problems that have no classical solution. In this paper we
introduce certain notions of non-Archimedean mathematics (in particular, of
nonstandard analysis) by means of an elementary topological approach; in
particular, we construct non-Archimedean extensions of the reals as appropriate
topological completions of $\mathbb{R}$. Our approach is based on the notion of
$\Lambda $-limit for real functions, and it is called $\Lambda $-theory. It can
be seen as a topological generalization of the $\alpha $-theory presented in
\cite{BDN2003}, and as an alternative topological presentation of the
ultrapower construction of nonstandard extensions (in the sense of
\cite{keisler}). To motivate the use of $\Lambda $-theory for applications we
show how to use it to solve a minimization problem of calculus of variations
(that does not have classical solutions) by means of a particular family of
generalized functions, called ultrafunctions.
