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Measure in functional analysis From Wikipedia, the free encyclopedia
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]
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
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