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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.[1]
Definition
Let be a family of Banach spaces, where may have arbitrarily large cardinality. Set
the product vector space.
The index set becomes a measure space when endowed with its counting measure (which we shall denote by ), and each element induces a function
Thus, we may define a function
and we then set
together with the norm
The result is a normed Banach space, and this is precisely the Lp sum of
Properties
Whenever infinitely many of the contain a nonzero element, the topology induced by the above norm is strictly in between product and box topology.
Whenever infinitely many of the contain a nonzero element, the Lp sum is neither a product nor a coproduct.
References
^Helemskii, A. Ya. (2006). Lectures and Exercises on Functional Analysis. Translations of Mathematical Monographs. American Mathematical Society. ISBN0-8218-4098-3.