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 V of y₀ there is a neighborhood U of x₀ such that if x ∈ U, then f(x) ∈ V.
$ Metric space def: The function f : X → Y tends to the limit y₀ as x → x₀ if for any positive ε there is a δ such that if d(x, x₀) < δ then d(f(x), y₀) < ε.
$ 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 → ∞.
> Online resources: see Wikipedia pages on limits of functions and limits of sequences.

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

Direct or Inductive Limit > s.a. lie group; uniform space.
$ 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^∞ X_i, with the topology X ⊃ A ∈ T iff A ∩ X_i ∈ T_i, for all i.
* Example: \(\mathbb C\)P^∞ is the inductive limit of \(\mathbb C\)P → \(\mathbb C\)P → ···.
$ Of C*-algebras: Given an inductive family (I, {\(\cal A\)_i}, {φ_ij}), its inductive limit is the set of equivalence classes of "Cauchy sequences", \(\cal A\)_∞:= { {a_i}_{i ∈ I} | a_i ∈ \(\cal A\)_i}, with || φ_ij(a_i) − a_j ||_m → 0 as i, j → ∞.
* Norm: || a ||_∞:= lim_{i → ∞} || a_i ||_i, for any representative family.
$ Of posets: Given (I, {P_i}, {φ_ij}), define P_∞:= ∪_{i ∈ I} P_i, with p <_∞ 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.
@ Poset completion: Meyer & Sorkin pr(89); in Bombelli & Meyer PLA(89); > s.a. the set of posets.
> Online resources: see Encyclopedia of Mathematics page; PlanetMath page; Wikipedia page.

Related Topics > see lorentzian geometry [limits of spacetimes]; projective limits.

main page – abbreviations – journals – comments – other sites – acknowledgements
send feedback and suggestions to bombelli at olemiss.edu – modified 8 sep 2019