Lp sum

Last updated November 05, 2020

In mathematics, and specifically in functional analysis, the L^p sum of a family of Banach spaces is a way of turning a subset of the product set of the members of the family into a Banach space in its own right. The construction is motivated by the classical L^p spaces.^[1]

Definition

Let $(X_{i})_{i\in I}$ be a family of Banach spaces, where $I$ may have arbitrarily large cardinality. Set

P:=\prod _{i\in I}X_{i},

the product vector space.

The index set $I$ becomes a measure space when endowed with its counting measure (which we shall denote by $\mu$ ), and each element $(x_{i})_{i\in I}\in P$ induces a function

I\to \mathbb {R} ,i\mapsto \|x_{i}\|.

Thus, we may define a function

\Phi :P\to \mathbb {R} \cup \{\infty \},(x_{i})_{i\in I}\mapsto \int _{I}\|x_{i}\|^{p}\,d\mu (i)

and we then set

\sideset {}{^{p}}\bigoplus \limits _{i\in I}X_{i}:=\{(x_{i})_{i\in I}\in P\mid \Phi ((x_{i})_{i\in I})<\infty \}

together with the norm

\|(x_{i})_{i\in I}\|:=\left(\int _{i\in I}\|x_{i}\|^{p}\,d\mu (i)\right)^{1/p}.

The result is a normed Banach space, and this is precisely the L^p sum of $(X_{i})_{i\in I}$ .

Properties

Whenever infinitely many of the $X_{i}$ contain a nonzero element, the topology induced by the above norm is strictly in between product and box topology.
Whenever infinitely many of the $X_{i}$ contain a nonzero element, the L^p sum is neither a product nor a coproduct.

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References

↑ Helemskii, A. Ya. (2006). Lectures and Exercises on Functional Analysis. Translations of Mathematical Monographs. American Mathematical Society. ISBN 0-8218-4098-3.

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[helemskii-1] Helemskii, A. Ya. (2006). Lectures and Exercises on Functional Analysis. Translations of Mathematical Monographs. American Mathematical Society. ISBN 0-8218-4098-3.

v t e Functional analysis (topics – glossary)
Spaces	Hilbert space Banach space Fréchet space topological vector space
Theorems	Hahn–Banach theorem closed graph theorem uniform boundedness principle Kakutani fixed-point theorem Krein–Milman theorem min-max theorem Gelfand–Naimark theorem Banach–Alaoglu theorem
Operators	bounded operator compact operator adjoint operator unitary operator Hilbert–Schmidt operator trace class unbounded operator
Algebras	Banach algebra C-algebra spectrum of a C-algebra operator algebra group algebra of a locally compact group von Neumann algebra
Open problems	invariant subspace problem Mahler's conjecture
Applications	Besov space Hardy space spectral theory of ordinary differential equations heat kernel index theorem calculus of variation functional calculus integral operator Jones polynomial topological quantum field theory noncommutative geometry Riemann hypothesis
Advanced topics	locally convex space approximation property balanced set Schwartz space weak topology barrelled space Banach–Mazur distance Tomita–Takesaki theory

Lp sum

Contents

Definition

Properties

Related Research Articles

References