analysis (differential/integral calculus, functional analysis, topology)
metric space, normed vector space
open ball, open subset, neighbourhood
convergence, limit of a sequence
compactness, sequential compactness
continuous metric space valued function on compact metric space is uniformly continuous
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symmetric monoidal (∞,1)-category of spectra
An infinite product is a sequence of numbers (usually real or complex written as . Like an infinite series, we are interested in knowing whether such a product converges, and if so, what it converges to.
For any group or semigroup , we can inductively define the finitary product -action
as
for all , such that currying the action results in the infinite product operator
over the function -module .
In a cartesian closed category , let be an multiplicative magma object, let be a natural numbers object in , and let be an infinite sequence object of elements in . Then there exists an infinite sequence object called the left infinite product object of inductively defined by and , and an infinite sequence object called the right infinite product object of inductively defined by and . The element is called the left -th partial product of the sequence , and the element is called the right -th partial product of the sequence . If the magma object is commutative and associative, then the left and right infinite product objects of are equal and just called an infinite product object of , where the element is the -th partial product.
A naive definition of convergence, by analogy with the sum of a series, would be that if the latter limit exists. However, this has the flaw that it could happen that this limit exists and yet might not, whereas we would like to be able to say that if converges then (by analogy with the fact that if converges then ). This failure can happen for two reasons:
If some , then since the partial products are eventually , regardless of the eventual behavior of the sequence .
If , then , whereas the sequence might approach any limit of absolute value or have no limit at all.
To avoid these “pathological” situations, we make the following modified definition.
Suppose at most finitely many of the are zero. We say that converges if
exists and is nonzero. If this is the case, we say that
If the above limit equals 0, one sometimes says that diverges to 0.
Last revised on May 28, 2021 at 22:07:51. See the history of this page for a list of all contributions to it.