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Let A = span{1, h^k|k ∈ N ∪ {0}} where h : [0, 1] → R is any

strictly monotone continuous function}. (a) Show that A is a dense
subalgebra of C([0, 1]) with respect to the uniform norm

1 Answer

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Final answer:

The question is about proving that a certain span of functions is a dense subalgebra of continuous functions on the interval [0, 1]. The set contains powers of a strictly monotone continuous function. To show denseness, one would typically refer to the Weierstrass approximation theorem.

Step-by-step explanation:

The student is asking about a concept in functional analysis, specifically about the properties of a subalgebra of the space of continuous functions, C([0, 1]), with respect to the uniform norm. The subalgebra A is defined to be the span of the set {1, h^k|k ∈ N ∪ {0}} where h is a strictly monotone continuous function on the interval [0, 1].

To show that A is a dense subalgebra, one would usually demonstrate two things: first, that it is indeed a subalgebra, meaning that it is closed under function addition, scalar multiplication, and multiplication; and second, that it is dense in the continuous functions on [0, 1], implying that for any continuous function f and any ε > 0, there exists a function g in A such that the uniform norm ||f-g|| < ε. Given that h is strictly monotone and continuous, its powers would also be continuous, and thus we can generate a vast array of continuous functions through linear combinations of these powers, which would approximate any continuous function due to the Weierstrass approximation theorem.

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