# properties of the Stone-Weierstrass Theorem

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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#### Solution Preview

See the attachment, please.

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 ...

#### Solution Summary

Properties of the Stone-Weierstrass Theorem are emphasized.