Lp sum

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 Lp 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.

References

↑ A. Ya. Helemskii (2006). "Lectures and Exercises on Functional Analysis". "Translations of Mathematical Monographs". American Mathematical Society.

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