Direct sum of topological groups explained

is called the topological direct sum^[1] of two subgroups

H₁

and

H₂

if the map

\beginH_1\times H_2 &\longrightarrow G \\(h_1,h_2) &\longmapsto h_1 h_2\end

is a topological isomorphism, meaning that it is a homeomorphism and a group isomorphism.

Definition

More generally,

is called the direct sum of a finite set of subgroups

H_1,\ldots,H_n

of the map

\begin\prod^n_ H_i &\longrightarrow G \\(h_i)_ &\longmapsto h_1 h_2 \cdots h_n\end

is a topological isomorphism.

If a topological group

is the topological direct sum of the family of subgroups

H_1,\ldots,H_n

then in particular, as an abstract group (without topology) it is also the direct sum (in the usual way) of the family

H_i.

Given a topological group

we say that a subgroup

is a topological direct summand of

(or that splits topologically from

) if and only if there exist another subgroup

K\leqG

such that

is the direct sum of the subgroups

and

A the subgroup

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

0 \to H\stackrel G\stackrel G/H\to 0

splits, where

is the natural inclusion and

\pi

is the natural projection.

Suppose that

as a subgroup. Then

is a topological direct summand of

The same assertion is true for the real numbers

^[2]

E. Hewitt and K. A. Ross, Abstract harmonic analysis. Vol. I, second edition, Grundlehren der Mathematischen Wissenschaften, 115, Springer, Berlin, 1979. MR0551496 (81k:43001)
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. MR0637201 (83h:22010)