Direct sum of topological groups

More generally, $G$ is called the direct sum of a finite set of subgroups $H\_1,\; \backslash ldots,\; H\_n$ of the map

$$\backslash begin\{align\}$$

\prod^n_{i=1} H_i &\longrightarrow G \\

(h_i)_{i\in I} &\longmapsto h_1 h_2 \cdots h_n

\end{align}

is a topological isomorphism.

If a topological group $G$ is the topological direct sum of the family of subgroups $H\_1,\; \backslash ldots,\; H\_n$ 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 $G,$ we say that a subgroup $H$ is a topological direct summand of $G$ (or that splits topologically from $G$) if and only if there exist another subgroup $K\; \backslash leq\; G$ 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

$$0\; \backslash to\; H\backslash stackrel\{i\}\{\{\}\; \backslash to\; \{\}\}\; G\backslash stackrel\{\backslash pi\}\{\{\}\; \backslash to\; \{\}\}\; G/H\backslash to\; 0$$

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

Suppose that $G$ is a locally compact abelian group that contains the unit circle $\backslash mathbb\{T\}$ as a subgroup. Then $\backslash mathbb\{T\}$ is a topological direct summand of $G.$ The same assertion is true for the real numbers $\backslash R$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).

• {{annotated link|Complemented subspace}}
• {{annotated link|Direct sum}}
• {{annotated link|Direct sum of modules}}

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Category:Topological groups

Category:Topology