Direct sum of topological groups

In mathematics, a topological group [math]\displaystyle{ G }[/math] is called the topological direct sum[1] of two subgroups [math]\displaystyle{ H_1 }[/math] and [math]\displaystyle{ H_2 }[/math] if the map [math]\displaystyle{ \begin{align} H_1\times H_2 &\longrightarrow G \\\ (h_1,h_2) &\longmapsto h_1 h_2 \end{align} }[/math] is a topological isomorphism, meaning that it is a homeomorphism and a group isomorphism.

## Contents

* 1 Definition
* 2 Topological direct summands
* 3 Examples
* 4 See also
* 5 References

## Definition

More generally, [math]\displaystyle{ G }[/math] is called the direct sum of a finite set of subgroups [math]\displaystyle{ H_1, \ldots, H_n }[/math] of the map [math]\displaystyle{ \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} }[/math] is a topological isomorphism.

If a topological group [math]\displaystyle{ G }[/math] is the topological direct sum of the family of subgroups [math]\displaystyle{ H_1, \ldots, H_n }[/math] then in particular, as an abstract group (without topology) it is also the direct sum (in the usual way) of the family [math]\displaystyle{ H_i. }[/math]

## Topological direct summands

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

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

## Examples

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

## See also

* Complemented subspace
* Direct sum - Operation in abstract algebra composing objects into "more complicated" objects
* Direct sum of modules - Operation in abstract algebra

## 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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Functional analysis (topics)

Topological vector spaces|

* Asplund
* Banach (list)
* Banach lattice
* Barrelled
* Bornological
* Brauner
* F-space
* Fréchet (tame)
* Hilbert (Inner product space
* Polarization identity)
* LF-space
* Locally convex (Seminorms/Minkowski functionals)
* Mackey
* Montel
* Nuclear
* Normed (norm)
* Quasinormed
* Reflexive
* Riesz
* Smith
* Stereotype
* Strictly convex
* Webbed
* Topological tensor product (of Hilbert spaces)

Topologies of function spaces|

* Dual
* Dual space (Dual norm)
* Operator
* Ultraweak
* Weak (polar
* operator)
* Mackey
* Strong (polar
* operator)
* Ultrastrong
* Uniform convergence

Linear operators|

* Adjoint
* Bilinear (form
* operator
* sesquilinear)
* (Un)Bounded
* Closed
* Compact (on Hilbert spaces)
* (Dis)Continuous
* Densely defined
* Fredholm
* Hilbert–Schmidt
* Functionals (positive)
* Normal
* Nuclear
* Self-adjoint
* Strictly singular
* Trace class
* Transpose
* Unitary

Operator theory|

* Banach algebras
* C*-algebras
* Spectrum (C*-algebra
* radius)
* Spectral theory (of ODEs
* Spectral theorem)
* Polar decomposition
* Singular value decomposition

Theorems|

* Banach–Alaoglu
* Banach–Mazur
* Banach–Saks
* Banach–Schauder (open mapping)
* Banach–Steinhaus (Uniform boundedness)
* Bessel's inequality
* Cauchy–Schwarz inequality
* Closed graph
* Closed range
* Eberlein–Šmulian
* Freudenthal spectral
* Gelfand–Mazur
* Gelfand–Naimark
* Goldstine
* Hahn–Banach (hyperplane separation)
* Kakutani fixed-point
* Krein–Milman
* Lomonosov's invariant subspace
* Mackey–Arens
* Mazur's lemma
* M. Riesz extension
* Riesz representation
* Parseval's identity
* Schauder fixed-point

Analysis|

* Abstract Wiener space
* Bochner space
* Differentiation in Fréchet spaces
* Derivatives (Fréchet
* Gateaux
* functional
* holomorphic)
* Integrals (Bochner
* Dunford
* Gelfand–Pettis
* regulated
* Paley–Wiener
* weak)
* Functional calculus (Borel
* continuous
* holomorphic)
* Inverse function theorem (Nash–Moser theorem)
* Measures (Lebesgue
* Projection-valued
* Vector)
* Weakly measurable function

Types of sets|

* Absolutely convex
* Absorbing
* Balanced
* Bounded
* Convex
* Convex cone (subset)
* Linear cone (subset)
* Radial
* Star-shaped
* Symmetric
* Zonotope

Subsets / set operations|

* Algebraic interior (core)
* Bounding points
* Convex hull
* Extreme point
* Interior
* Minkowski addition
* Polar

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Topological vector spaces (TVSs)

Basic concepts|

* Banach space
* Continuous linear operator
* Functionals
* Hilbert space
* Linear operators
* Locally convex space
* Homomorphism
* Topological vector space
* Vector space

Main results|

* Closed graph theorem
* F. Riesz's theorem
* Hahn–Banach (hyperplane separation
* Vector-Valued Hahn-Banach)
* Open mapping (Banach–Schauder) (Bounded inverse)
* Uniform boundedness (Banach–Steinhaus)

Maps|

* Almost open
* Bilinear (form
* operator) and Sesquilinear forms
* Closed
* Compact operator
* Continuous and Discontinuous Linear maps
* Densely defined
* Homomorphism
* Functionals
* Norm
* Operator
* Seminorm
* Sublinear
* Transpose

Types of sets|

* Absolutely convex/disk
* Absorbing/Radial
* Affine
* Balanced/Circled
* Banach disks
* Bounding points
* Bounded
* Complemented subspace
* Convex
* Convex cone (subset)
* Linear cone (subset)
* Extreme point
* Pre-compact/Totally bounded
* Radial
* Radially convex/Star-shaped
* Symmetric

Set operations|

* Affine hull
* (Relative) Algebraic interior (core)
* Convex hull
* Linear span
* Minkowski addition
* Polar
* (Quasi) Relative interior

Types of TVSs|

* Asplund
* B-complete/Ptak
* Banach
* (Countably) Barrelled
* (Ultra-) Bornological
* Brauner
* Complete
* (DF)-space
* Distinguished
* F-space
* Fréchet (tame Fréchet)
* Grothendieck
* Hilbert
* Infrabarreled
* Interpolation space
* LB-space
* LF-space
* Locally convex space
* Mackey
* (Pseudo)Metrizable
* Montel
* Quasibarrelled
* Quasi-complete
* Quasinormed
* (Polynomially
* Semi-) Reflexive
* Riesz
* Schwartz
* Semi-complete
* Smith
* Stereotype
* (B
* Strictly
* Uniformly convex
* (Quasi-) Ultrabarrelled
* Uniformly smooth
* Webbed
* With the Approximation property

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