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 $$I\to ℝ,i\mapsto \Vert x_i\Vert .$$

Thus, we may define a function $$\mathrm{\Phi }:P\to ℝ\cup \{\mathrm{\infty }\},(x_i{)}_{i\in I}\mapsto \underset{I}{\int }\Vert x_i{\Vert }^{p}d\mu (i)$$ and we then set $$⨁_{{}^p}{\mathrm{\backslash limits}}_{i\in I}{X}_i:=\{(x_i{)}_{i\in I}\in P\mid \mathrm{\Phi }((x_i{)}_{i\in I})<\mathrm{\infty }\}$$ together with the norm $$\Vert (x_i{)}_{i\in I}\Vert :={(\underset{i\in I}{\int }\Vert x_i{\Vert }^{p}d\mu (i))}^{1/p}.$$

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.