Direct sum of topological groups

In mathematics, a topological group G is called the topological direct sum[1] of two subgroups H1 and H2 if the map

\begin{align}
H_1\times H_1 &\longrightarrow G \\
(h_1,h_2)     &\longmapsto     h_1 h_2
\end{align}

is a topological isomorphism.

More generally, G is called the direct sum of a finite set of subgroups H_i, i=1,\ldots, n of the map

\begin{align}
\prod^n_{i=1} H_i&\longrightarrow G \\
(h_i)_{i\in I}    &\longmapsto     h_1 h_2 \cdots h_n
\end{align}

Note that if a topological group G is the topological direct sum of the family of subgroups H_i then in particular, as an abstract group (without topology) it is also the direct sum (in the usual way) of the family H_i .

Topological direct summands

Given a topological group G, we say that a subgroup H is a topological direct summand of G (or that splits topologically form G ) if and only if there exist another subgroup K  G such that G is the direct sum of the subgroups H and K.

A the subgroup H is a topological direct summand if and only if the extension of topological groups

0 \to H\stackrel{i}{{} \to {}} G\stackrel{\pi}{{} \to {}} G/H\to 0

splits, where i is the natural inclusion and \pi is the natural projection.

Examples

References

  1. E. Hewitt and K. A. Ross, Abstract harmonic analysis. Vol. I, second edition, Grundlehren der Mathematischen Wissenschaften, 115, Springer, Berlin, 1979. MR0551496 (81k:43001)
  2. Armacost, David L. The structure of locally compact abelian groups. Monographs and Textbooks in Pure and Applied Mathematics, 68. Marcel Dekker, Inc., New York, 1981. vii+154 pp. ISBN 0-8247-1507-1 MR0637201 (83h:22010)
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