Re: bounded variation



Andrew wrote:

> Have read various (all quite similar definitions) of bounded
> variation (eg http://en.wikipedia.org/wiki/Bounded_variation)
>
> im slightly confused by what it is
> do i do the sum first then take the sup? (but isnt that
> just going to be the same as if i didnt take the sup?)
>
> if i take the sup of each term? that doesnt seem to make
> sense to me either...
>
> anyone shed some light on this..

I only saw one definition of bounded variation at the web page
you gave, and it seems fine to me. As for which you do first,
note that the definition says the supremum is taken over
all partitions of [a,b]. Since the sum doesn't make sense
until a partition is chosen, it has to be that the supremum
comes last. It also doesn't make sense to take the supremum
before the sum because then you wouldn't be taking the
supremum of a collection of real numbers, and even if
you somehow managed to do this, there would be no point
to the sum since there would then only be one number
involved (the number that is the supremum).

You probably need to see some specific examples worked out
to clear things up. Some of these might help:

http://books.google.com/books?as_q=examples&as_epq=bounded+variation

In particular, try:

6'th google print result
p. 210 in "A First Course in Analysis" by George Pedrick

19'th google print result
p. 252 in "Mathematical Analysis 2edmetallury 2edse"

92'nd google print result
p. 245 in "Advanced Mathematical Methods" by Adam Ostaszewski

Dave L. Renfro

.