Question: I just wanted to clear this up, because i think a question I'm attempting may be somewhat badly written. It talks about a set of functions being 'bounded pointwise by 1' - but isn't that just 'bounded by 1'? I don't understand in what sense it is bounded pointwise by a single value, because surely as 1 is constant, it is uniformly bounded by 1? Perhaps the question is just confusingly phrased.

Thanks for the help - M.

Answer: 
"Bounded pointwise by 1" and "uniformly bounded by 1" mean exactly the same thing. Perhaps it is written this way for emphasis. You would have to ask the author.
If some norm were mentioned, the functions could be bounded in norm (and this does not necessarily mean pointwise bounded). Likely, however, the author is just repeating himself at the risk of being repetitive.