Question

Difference between equicontinuous and uniformly convergent

Answer 1

Explanation:

My definition is that:

Uniformly Equicontinuous: ∀ϵ>0,∃δ>0 such that |s−t|<δ and n∈N then |fn(t)−fn(s)|<ϵ

Uniformly continuous: ∀ϵ>0,∃δ>0 such that ∀s,t∈[a,b], |s−t|<δ and n∈N then |fn(t)−fn(s)|<ϵ

Answer 2

Answer:

Uniformly continuous :

If E,E′ are metric spaces then a function f:E→E′ is uniformly continuous if for each ε>0 there exists δ>0 such that sup{d′(f(x),f(y)):d(x,y)<δ}<ε.

Uniformly equicontinuous :

A family of functions (fα)α∈I from E to E′ is uniformly equicontinuous if for each ε>0 there exists δ>0 such that supα∈Isup{d′(fα(x),fα(y)):d(x,y)<δ}<ε.