Limits

In General > s.a. Filter; sequence.
$ Topological def: The function f : X → Y tends to the limit y₀ as x → x₀ if for any neighborhood...
$ Metric space def: The function f : X → Y tends to the limit y₀ as x → x₀ if for any ...
$ Heine limit: The function f : X → Y has y₀ as the Heine limit as x → x₀ if for any sequence {x_n} converging to x₀ in X, the sequence {f(x_n)} converges to y₀ in Y as n → .

Infimum and Supremum Limits
$ lim sup: Given a sequence {x_i}_{i in N}, define a_m:= l.u.b. {x_m, x_m+1, ...}; Then lim sup_{n to infty} x_n:= lim_{m to infty} a_m.
> Online resources: see Wikipedia page.

Direct or Inductive Limit > s.a. lie group.
$ Of topological spaces: Given the sequence X₁ X₂ ··· X_i ··· of topological spaces {(X_i, T_i)}, its inductive limit is the space X:= _i=1^infty X_i, with the topology X A T iff A X_i T_i, for all i.
* Example: CP^infty is the inductive limit of CP → CP → ···.
$ Of C*-algebras: Given an inductive family (I, {_i}, {_ij}), its inductive limit is the set of equivalence classes of "Cauchy sequences", _infty:= { {a_i}_{i in I} | a_{i i}}, with _ij(a_i) – a_{j m} → 0 as i, j → .
* Norm: a _infty:= lim_{i to infty} a_{i i}, for any representative family.
$ Of posets: Given (I, {P_i}, {_ij}), define P_infty:= _{i in I} P_i, with p <_infty q iff there exists i in I such that p, q P_i and p <_i q.
@ General references: in Eilenberg & Steenrod 52; Fell & Doran 88; Murphy 90.
@ Completion: Meyer & Sorkin pr; in Bombelli & Meyer PLA(89).

Related Topics > see projective limits.

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