1. Limited time only! Sign up for a free 30min personal tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Homework Help: Stone-Weierstrass theorem problem

  1. Dec 6, 2004 #1
    I'm working on a problem that has to do with the Stone-Weierstrass theorem. This is the problem:

    The way that I've been trying to do it is to produce an algebra of continuous functions that separates points and contains constant functions. If I define [itex]A[/itex] to be the set of all \sum_{i=1}^n g_ih_i where [itex]g_1,\ldots,g_n[/itex] are continuous on [itex]X[/itex] and [tex]h_1,\ldots,h_n[/tex] are continuous on [itex]Y[/itex], it is easy to show that constant multiples of functions in [itex]A[/itex] are in [itex]A[/itex], [itex]A[/itex] is closed under multiplication, [itex]A[/itex] separates points, and [itex]A[/itex] contains the constant functions. What I am having trouble showing is that [itex]A[/itex] is closed under addition (ie. that [itex]A[/itex] actually is an algebra). Is this true? If it is not then does anybody know of a way to come up with an algebra for this problem so that I could apply Stone-Weierstrass? Any help would be greatly appreciated.
  2. jcsd
  3. Dec 6, 2004 #2
    Okay now I feel stupid. Now thinking about it sums are included basically by definition.
Share this great discussion with others via Reddit, Google+, Twitter, or Facebook