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 H1×H2⟶G(h1,h2)⟼h1h2 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, G is called the direct sum of a finite set of subgroups H1,…,Hn of the map ∏i=1nHi⟶G(hi)i∈I⟼h1h2⋯hn is a topological isomorphism.

If a topological group G is the topological direct sum of the family of subgroups H1,…,Hn then in particular, as an abstract group (without topology) it is also the direct sum (in the usual way) of the family Hi.

## 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 from 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->H->iG->πG/H->0 splits, where i is the natural inclusion and π is the natural projection.

## Examples

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

## See also

* Complemented subspace
* Direct sum - Algebraic structure formed from a collection of algebraic structures
* 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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* t
* e

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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