What is the meaning of tending uniformly to infinity in Harnack's Principle?

[redrzewski]

Ahlfors version of this theorem says that a sequence of harmonic functions {Un} tends UNIFORMLY to infinity on compact subsets, or tends to a harmonic limit function uniformly on compact sets.

Can someone please clarify what tending uniformly to infinity means?

In particular, it seems like a set of harmonic {Un} where Uk = k (such that each function is constant) tends non-uniformly to infinity.

So I must be missing something somewhere.

thanks

 

[jostpuur]

f_n:K\to\mathbb{C} (n=1,2,3\ldots) converges towards infinity uniformly, if for all R>0 there exists N\in\mathbb{N} such that |f_n(z)|>R for all z\in K and n\geq N.

Simply extend [0,\infty[ to [0,\infty] (with topology homeomorphic with [0,1]), and threat \infty as a constant so that a sequence of functions can converge uniformly towards the corresponding constant function.

In point-wise convergence to infinity would mean that for each z\in K and R>0 there exists N\in\mathbb{N} such that |f_n(z)|>R for all n\geq N.

Constant functions surely converge uniformly if they converge somehow.

 

[redrzewski]

Thanks. That definition clears things up greatly.