Direct sum of topological groups

In mathematics, a topological group ${\displaystyle G}$ is called the topological direct sum^[1] of two subgroups ${\displaystyle H_{1}}$ and ${\displaystyle H_{2}}$ if the map $${\displaystyle {\begin{aligned}H_{1}\times H_{2}&\longrightarrow G\\(h_{1},h_{2})&\longmapsto h_{1}h_{2}\end{aligned}}}$$ is a topological isomorphism, meaning that it is a homeomorphism and a group isomorphism.

More generally, ${\displaystyle G}$ is called the direct sum of a finite set of subgroups ${\displaystyle H_{1},\ldots ,H_{n}}$ of the map $${\displaystyle {\begin{aligned}\prod _{i=1}^{n}H_{i}&\longrightarrow G\\(h_{i})_{i\in I}&\longmapsto h_{1}h_{2}\cdots h_{n}\end{aligned}}}$$ is a topological isomorphism.

If a topological group ${\displaystyle G}$ is the topological direct sum of the family of subgroups ${\displaystyle H_{1},\ldots ,H_{n}}$ then in particular, as an abstract group (without topology) it is also the direct sum (in the usual way) of the family ${\displaystyle H_{i}.}$

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

A the subgroup ${\displaystyle H}$ is a topological direct summand if and only if the extension of topological groups $${\displaystyle 0\to H{\stackrel {i}{{}\to {}}}G{\stackrel {\pi }{{}\to {}}}G/H\to 0}$$ splits, where ${\displaystyle i}$ is the natural inclusion and ${\displaystyle \pi }$ is the natural projection.

Suppose that ${\displaystyle G}$ is a locally compact abelian group that contains the unit circle ${\displaystyle \mathbb {T} }$ as a subgroup. Then ${\displaystyle \mathbb {T} }$ is a topological direct summand of ${\displaystyle G.}$ The same assertion is true for the real numbers ${\displaystyle \mathbb {R} }$^[2]

• Complemented subspace
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