Lp sum

In mathematics, and specifically in functional analysis, the Lp 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.

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 $$I \to \Reals, i \mapsto \|x_i\|.$$

Thus, we may define a function $$\Phi: P \to \Reals \cup \{\infty\}, (x_i)_{i \in I} \mapsto \int_I \|x_i\|^p \,d \mu(i)$$ 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 Lp 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 Lp sum is neither a product nor a coproduct.