# hmwk9 - MATH 202A HOMEWORK 9 The following assignment is...

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Unformatted text preview: MATH 202A HOMEWORK 9 The following assignment is due Wed., Oct. 31. Problem 1 . (a) (Royden problem #43). Let A be an algebra of continuous real- valued functions on a compact space X and assume that A separates the points of X . Then either ¯ A = C ( X ) or there is a point p ∈ X and ¯ A = { f ∈ C ( X ) | f ( p ) = 0 } . Hint: If 1 ∈ ¯ A , we are done. If ∃ f ∈ A , which is nowhere zero, then 1 ∈ ¯ A (see my extra hint at the end of the assignment). If for each x ∈ X , there is an f ∈ A with f ( x ) 6 = 0, then ∃ g ∈ A with g > 0 everywhere.] (b) Prove that the set of smooth functions with compact support C ∞ c ( R ) is uniformly dense in the space of continuous functions that vanish at infinity C ( R ). Here, a smooth function is one that is infinitely differentiable; the support of a function f is the closure of the set { x ∈ R | f ( x ) 6 = 0 } , and the space of functions C ( R ) is the set of all continuous functions f ( x ) on R such that for each > 0 there exists a compact set K ⊂ R such that | f ( x ) | < on K c . Suggestion: Identify C ( R ) with the subspace of C ( S 1 ) of functions vanishing at a point, and apply (a). You will need to build one smooth function with compact support. One way to do this is to show φ ( x ) = e- 1 /x for x > for x ≤ is smooth and define ψ ( x ) = φ (1- x ) φ (1 + x )....
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hmwk9 - MATH 202A HOMEWORK 9 The following assignment is...

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