Uniform vs pointwise convergence

[dimitri151]

I was reading Royden when I came across this cryptic statement:pg 222, "The concept of uniform convergence of a sequence of functions is a metric concept. The concept of pointwise convergence is not a metric concept." Can anyone illuminate this?

 

[lavinia]

I was reading Royden when I came across this cryptic statement:pg 222, "The concept of uniform convergence of a sequence of functions is a metric concept. The concept of pointwise convergence is not a metric concept." Can anyone illuminate this?

Uniform convergence is defined with measurements of the distance between functions.

|f(x) - f_n(x)| < a for all n large enough and for any given number, a. Here the inequality is saying that the distance between the function,f anf f_n is less than a. This is what is meant by saying that uniform convergence is a metrical concept. It uses a metric on the space of functions. Generally, there is a continuum of metrics on measurable functions, one for each real number greater than or equal to 1. The uniform metric is also called the L-infinity metric.

Pointwise convergence does not use any idea of a metric on the functions. All that it requires is that for each point the values of the functions in the sequence converge.

Uniform convergence is used to prove the existence of extremely strange functions, for instance space filling curves, continuous curves that can completely fill a region such as a square or a cube. I think - though I am not sure - when this was first discovered in the 19'th century, people worried that 2 and 3 dimensions were really just curves wrapped up in a wild way and that all dimensions were therefore really 1 dimensional.

 

[HallsofIvy]

The usual "Calculus" definition of convergence of a sequence of functions is
"$\{f_n(x)\}$ converges to f(x) if and only if, for each $x_0$, given $\epsilon> 0$, there exist an integer N such that if n>N then $|f_n(x_0)- f(x_0)|< \epsilon$"
which looks like a metric statement.

However, you can phrase it more generally as
"$\{f_n(x)\}$ converges to f(x) if and only if, given an open set U containing f(x_0), there exist an integer N such that if n> N then $f_n(x_0)\in U$" which can be given in any topological space, not just metric spaces.

But the definition of "uniform" convergence of a sequence of functions is
"$\{f_n(x)\}$ converges to f(x) if and only if, for each $x_0$, given $\epsilon> 0$, there exist an integer N such that if n>N then $|f_n(x_0)- f(x_0)|< \epsilon$"

Do you see the difference? The "for each $x_$" and "given $\epsilon> 0$" have been switched. That means that, for a given $\epsilon$ the same N must work for every $x_0$. Comparing it to the second definition, above, that did not require a metric, it is saying that the open sets, at different $f(x_0)$ must be of the same "size"- and "size" of sets is only defined in metric spaces.

 

[dimitri151]

But the definition of "uniform" convergence of a sequence of functions is
"$\{f_n(x)\}$ converges to f(x) if and only if, for each $x_0$, given $\epsilon> 0$, there exist an integer N such that if n>N then $|f_n(x_0)- f(x_0)|< \epsilon$"

For uniform convergence isn't there one N for all x rather than an N for each x?

 

[Bacle]

Actually, I think uniform convergence can be defined in uniform spaces, even non-metrizable ones.

 

[snipez90]

For uniform convergence isn't there one N for all x rather than an N for each x?

Yes.

Actually, I think uniform convergence can be defined in uniform spaces, even non-metrizable ones.

Well yeah, that seems to be the whole point. But this avenue of generalization seems to be mostly of interest to topologists. Royden's point, as lavinia hinted at, is that in analysis we care about the distance between functions. Many mathematicians in the early 19th century thought that pointwise convergence captured this idea, but the notion of uniform convergence was what was needed.