'less than or approximately' holds uniformly over set?

Question

I was reading these notes on regularity structures, and came across this (equation 2.5)

There exists a linear map such that $$\mbox{left hand side } \lesssim \mbox{ right hand side}$$ locally uniformly over $x$.

First: what exactly does $\lesssim$ mean?

Second, what does it mean for a bound to hold locally uniformly over $x$? Does it mean that the left hand side converges locally uniformly to the right hand side for every $x$?

Answer 1

$f(x)\lesssim g(x)$ if $f(x)\le C g(x)$

"locally uniformly bounded" means what it says. This bound holds uniformly, at least locally.