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.

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.