An Introduction to the Geometry of Numbers pp 121-174 | Cite as

# MAHLER’S compactness theorem

Chapter

## Abstract

So far we have been concerned with one lattice at a time. In this chapter we are concerned with properties of sets of lattices. We first must define what is meant by two lattices Λ and
are near those of the identity transformation, that is if

**M**being near to each other; and this is done by means of homogeneous linear transformations. A homogeneous linear transformation*=***X***of***τx***n*-dimensional euclidean space into itself is said to be near to identity transformation if the coefficients τ_{ ij }in$$ {X_i} = \sum\limits_{1\underline \le \,i\,\underline \le \,n} {{\tau _{ij}}{x_j}} \,\,\left( {1\,\underline \le \,i\,\underline \le \,n} \right) $$

$$ \left| {{\tau _{ii}} - \,1} \right|\,\,\,\left( {1\,\,\underline \le \,i\,\underline \le \,n} \right) $$

and
are all small.

$$ \left| {{\tau _{ij}}} \right|\,\,\,\,\left( {1\,\,\underline \le \,i\,\underline \le \,n,1\,\,\underline \le \,j\,\underline \le \,n,\,i \ne \,j\,} \right) $$

### Keywords

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## Copyright information

© Springer-Verlag Berlin Heidelberg 1997