> > Hatto von Aquitanien wrote: >> Gene Ward Smith wrote: > >> > Consider all the continuous real-valued functions on the interval [0, >> > 1]. If we define f+g by >> > (f+g)(x) = f(x)+g(x), and k*f by (k*f)(x) = k*f(x), these satisfy the >> > axioms for a real vector space. >> >> In R^n? n is finite in this discussion. It may not have been stated >> explicitly, but it should be clear from the context upon review. > > What rubbish; there was no such implication.
Curious. Perhaps this is another one of those cultural differences. I have always taken {1,2,...,n} to designate a finite range of positive integers, and therefore, by implication R^n to be a set of n-tuples with a finite number of components. In order for me to consider the possibility that n tends to infinity, it would have to be stated explicitly, or be obvious from the context. So, let me be clear. Edwards was fairly specific about this being a finite dimensional space.
> However, we can do > something similar for finite dimensions. Consider the real vector space > of all real-analytic functions on R. Take the kernel of the derivative > operator applied thrice: that is, take all the real-analytic functions > f for which D^3(f) is the zero function. There's your finite > dimensional space. > > If you don't like that one, how about this: all functions of the form A > sin(x + B) for A and B any real number.
