“Lp sum”的意思、由来-开放百科全书

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Lp sum

释义

Definition
Properties
References

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 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 L^p 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 L^p sum is neither a product nor a coproduct.

References

^[1]

1 : Banach spaces

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