[user070223]

I've skimmed the result couple of weeks ago, and if I remember it states that there from all machines which takes t time to accomplish something there is one such that sqrt(t) memory is used. so it's at most but not on avg nor amorotized[not sure if et even make sense it the space where the problem lies] (you take the least memory hungry machine)

[cma]

There must be more constraints, what if the problem is copying a string? Or is it only for turing tape machines where it has to backtrack during the copy, increasing number of steps?

[jasperry]

That's a good example! So the sqrt(n) space in the result must refer to space beyond that needed to write down the output--basically a working memory scratchpad. In which case, a string-copying function could use just a constant amount of scratch space.

I mean, for all I know, the result may have been proved in the model of decision problems where the output is always one bit. But I'd guess that it generalizes just fine to the case where you have to write longer outputs.

However, your question does make me wonder if the result still applies in the RAM model, where you can move to any spot on the tape in constant time. My theory knowledge is getting rusty...