L^p 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 L^p spaces.[1]

Definition

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

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

the product vector space.

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

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

Thus, we may define a function

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

and we then set

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

together with the norm

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

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

Properties

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

References

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