Lecture Notes

Uniform convergence · {date} · {course}

Pointwise convergence

The sequence (ƒ_n) converges pointwise to ƒ on E if, for every x∈E and every ε>0, there is N(x,ε) such that

|(ƒ_n)(x)−ƒ(x)|<ε for all n>N(x,ε)

The index N may depend on x, which makes pointwise convergence weak.

Uniform convergence

(ƒ_n)→ƒ uniformly on E if for every ε>0 there is N(ε), the same N for every x with

(sup_x∈E)|(ƒ_n)(x)-ƒ(x)|<ε for all n>N

Equivalently ‖(ƒ_n)-ƒ‖_∞→0.

The example that separates them

(ƒ_n)(x)=x^n on [0,1]

Pointwise the limit is

ƒ(x)={[0,0≤x<1],[1,x=1])

But,

(sup_x∈[0,1])|(ƒ_n)(x)-ƒ(x)|=(sup_x∈[0,1))(x^n)=1for every n

so the sup never goes to zero and convergence is not uniform. Each (ƒ_n)​ is continuous, the limit is not, and that's the whole point: pointwise convergence does not preserve continuity. The failure is entirely at the right endpoint.

On [0,a] with a<1, the supremum is a^n→0, so it is uniform there.

Theorem

If (ƒ_n) is continuous on E for each nand (ƒ_n)→ƒ uniformly on E, then ƒ is continuous on E.

Proof. Fix (x_0)∈E and ε>0. Choose n with ‖(ƒ_n)-ƒ‖_∞<ε/3. Since (ƒ_n)is continuous at (x_0), choose δ>0 so that |x−(x_0)|<δ gives |(ƒ_n)(x)−(ƒ_n)((x_0))|<ε/3. Then

|ƒ(x)−ƒ((x_0))|≤|ƒ(x)−(ƒ_n)(x)|+|(ƒ_n)(x)−(ƒ_n)((x_0))|+|(ƒ_n)((x_0))−ƒ((x_0))|<ε/3+ε/3+ε/3<ε

The middle term needs continuity of (ƒ_n). The outer two need uniformity — with only pointwise convergence the first term's N would depend on x, and x is still moving.

Weierstrass M-test

If |(u_n)(x)|≤(M_n) for all x∈E and (∑_^)((M_n))<∞, then (∑_^)((u_n)) converges uniformly on E.

Applied to (∑_n=1^∞)((sin(n)*x)/(n^2)):

|(sin(n)*x)/(n^2)|≤1/(n^2)=(M_n),(∑_n=1^∞)(1/(n^2))=(π^2)/6<∞

So the series converges uniformly on all of ℝ, and by the theorem above its sum is continuous everywhere.

Where it fails. For (∑_^)((sin(n)*x)/n) the bound gives (M_n)=1/n and (∑_^)(1)/n diverges, so the M-test says nothing. The series does still converge for every x, just not by this argument — the test is sufficient, not necessary.

Loose end

{Uniform convergence lets you swap (lim_)() and (∫_^)() on a bounded interval. Does it need the interval to be closed?}

Next lecture. Equicontinuity, Arzelà–Ascoli.