Limits |

**In General** > s.a. Filter; sequence.

$ __Topological def__: The
function *f *: *X* → *Y* tends
to the limit *y*_{0} as *x* →
*x*_{0} if for any neighborhood *V* of *y*_{0} there is a neighborhood *U* of *x*_{0} such that if *x* ∈ *U*, then *f*(*x*) ∈ *V*.

$ __Metric space def__: The
function *f* : *X* → *Y* tends
to the limit *y*_{0} as *x* → *x*_{0} if
for any positive *ε* there is a *δ* such that if *d*(*x*,* x*_{0}) < *δ* then *d*(*f*(*x*), *y*_{0}) < *ε*.

$ __Heine limit__: The function *f* : *X* → *Y* has *y*_{0}
as the Heine limit as *x* → *x*_{0} if
for any sequence {*x*_{n}}
converging to *x*_{0} in *X*, the sequence
{*f*(*x*_{n})} converges
to *y*_{0} in *Y* as *n* → ∞.

> __Online resources__:
see Wikipedia page.

**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 *X*_{1} ⊂ *X*_{2} ⊂ ··· ⊂ *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.

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send feedback and suggestions to bombelli at olemiss.edu – modified 14
feb 2016