Sunday, August 2, 2009

Infimum Jest

For the math lovers out there, this post is for you. In the footnote to a recent post ("A Visitor from Redmond") I referred to an infimum function. From Wolfram Math World1, here's a definition of infimum (Hat tip to M.I., my finance consultant):


The infimum is the greatest lower bound of a set S, defined as a quantity m such that no member of the set is less than m, but if epsilon is any positive quantity, however small, there is always one member that is less than m+epsilon (Jeffreys and Jeffreys 1988). When it exists (which is not required by this definition, e.g., infR does not exist), the infimum is denoted infS or inf_(x in S)x. The infimum is implemented in Mathematica as MinValue[f, constr, vars].

Consider the real numbers with their usual order. Then for any set M subset= R, the infimum infM exists (in R) if and only if M is bounded from below and nonempty.

More formally, the infimum infS for S a (nonempty) subset of the affinely extended real numbers R^_=R union {+/-infty} is the largest value y in R^_ such that for all x in S we have x>=y. Using this definition, infS always exists and, in particular, infR=-infty.

Whenever an infimum exists, its value is unique.

SEE ALSO: Infimum Limit, Lower Bound, Supremum

Note that Blogger converted some of the symbols in that definition to text, e.g., writing out the Greek letter epsilon instead of showing the Greek symbol for it.

Had I known of this back in December, when I wrote this post, "Infinite Connections or Infinite Jest", I could have used "Infimum Connections" as an alternative, obscurer post title.

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