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# properties of the Stone-Weierstrass Theorem

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Let be a compact interval and let A be a collection of continuous functions on which satisfy the properties of the Stone-Weierstrass Theorem
[Stone-Weierstrass Theorem: Let K be a compact subset of and let A be a collection of continuous functions on K to R with the properties:
a) The constant function belongs to A.
b) If
c) if
d) if
Then any continuous function K to R can be uniformly approximated on K by functions in A. ]

Show that any continuous function on to R can be uniformly approximated by functions of the form where .

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Let be a compact interval and let A be a collection of continuous functions on which satisfy the properties of the Stone-Weierstrass Theorem
[Stone-Weierstrass Theorem: Let K be a compact subset of and let A be a collection of continuous functions on K to R with the properties:
a) The constant function belongs to A.
b) If
c) if
d) if
Then any continuous function K to R can be uniformly approximated on K by functions in A. ]

Show that any continuous function on to R can be uniformly approximated by functions of the form where .
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Since is compact, we just need to show that the collection of functions generated by the set , that is, the collection of all linear conbinations of the form , satisfies the conditions a) - d)above.

Show that if and are continuous at and respectively, then is also continuous at .

First, notice, that if is continuous at , then the function is continuous at for any

Indeed, is continuous at means that for every epsilon bigger than zero there's a delta bigger than zero such that if , then

is continuous at means that for every epsilon bigger than zero there's a delta such that if then

Given , find such that as soon as , we have
Now, let be a point such that . Then we have:
Since , we have , and so

Similarly, we can show that if is continuous at , then is continuous at for any

Now, represent and use the fact that the product of contunious function is continuous.

Now, let's check the conditions a) - d) for the elements of :
a) Since the constant function belongs to A, we have that the constant function belongs to .
b) Since is defined as the set of all linear combinations of functions of the form
, we have this condition fulfilled automatically.
c) It's enough to check that if and ,
then , and
by the properties of we have that both and belong to .
d) Given two points . We need to show that there's a function in such that .

If then either or (or both).
Suppose, .
Since A satisfies the hypothesis of Stone-Weierstrass theorem, there's a function f in A such that . Consider the function where e(y) is the constant function . Then, .
If then we must have . Find a function such that and define . We'll have .

Thus, the conditions of Stone-Weierstrass Theorem are fulfilled for the collection , and we conclude that every fontinuous function can be uniformly approximated by the elements of , that is, the functions of the form where

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