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 L^p spaces.^[1]

Definition

Let [math]\displaystyle{ (X_i)_{i \in I} }[/math] be a family of Banach spaces, where [math]\displaystyle{ I }[/math] may have arbitrarily large cardinality. Set [math]\displaystyle{ P := \prod_{i \in I} X_i, }[/math] the product vector space.

The index set [math]\displaystyle{ I }[/math] becomes a measure space when endowed with its counting measure (which we shall denote by [math]\displaystyle{ \mu }[/math]), and each element [math]\displaystyle{ (x_i)_{i \in I} \in P }[/math] induces a function [math]\displaystyle{ I \to \Reals, i \mapsto \|x_i\|. }[/math]

Thus, we may define a function [math]\displaystyle{ \Phi: P \to \Reals \cup \{\infty\}, (x_i)_{i \in I} \mapsto \int_I \|x_i\|^p \,d \mu(i) }[/math] and we then set [math]\displaystyle{ \sideset{}{^p}\bigoplus\limits_{i\in I} X_i := \{ (x_i)_{i \in I} \in P \mid \Phi((x_i)_{i \in I}) \lt \infty\} }[/math] together with the norm [math]\displaystyle{ \|(x_i)_{i \in I}\| := \left( \int_{i \in I} \|x_i\|^p \, d\mu(i) \right)^{1/p}. }[/math]

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

Properties

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

References

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