5. Suppose fn converges to uniformly to f on [0, 1], all of the fn and f are differentiable withcontinuous derivatives, and that fh converges uniformly to g. Show that f' = g. HINT: TheFundamental Theorem of Calculus says that f(x) = So f'(t)dt + f(0).6. Let f be a continuous function on a compact metric space X and form the subalgebraA = span{1, f, f², f³, ..., f", ...}in C(X). Using Stone's theorem, show that A is dense in C(X) if ƒ is injective.

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5. Suppose fn converges to uniformly to f on [0, 1), all of the fn and f are differentiable with continuous derivatives, and that fh converges uniformly to g. Show that f' = g. HINT: The Fundamental Theorem of Calculus says that f(x) = S f'(t)dt + f(0). 6. Let ƒ be a continuous function on a compact metric space X and form the subalgebra A = span{1, f, f², f³,.. .., f",...} in C(X). Using Stone's theorem, show that A is dense in C(X) if ƒ is injective.

5. Suppose fn converges to uniformly to f on [0, 1), all of the fn and f are differentiable with continuous derivatives, and that fh converges uniformly to g. Show that f' = g. HINT: The Fundamental Theorem of Calculus says that f(x) = S f'(t)dt + f(0). 6. Let ƒ be a continuous function on a compact metric space X and form the subalgebra A = span{1, f, f², f³,.. .., f",...} in C(X). Using Stone's theorem, show that A is dense in C(X) if ƒ is injective.

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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Concept explainers

Rate of Change

The relation between two quantities which displays how much greater one quantity is than another is called ratio.

Slope

The change in the vertical distances is known as the rise and the change in the horizontal distances is known as the run. So, the rise divided by run is nothing but a slope value. It is calculated with simple algebraic equations as:

Question

5. Suppose fn converges to uniformly to f on [0, 1], all of the fn and f are differentiable with
continuous derivatives, and that fh converges uniformly to g. Show that f' = g. HINT: The
Fundamental Theorem of Calculus says that f(x) = So f'(t)dt + f(0).
6. Let f be a continuous function on a compact metric space X and form the subalgebra
A = span{1, f, f², f³, ..., f", ...}
in C(X). Using Stone's theorem, show that A is dense in C(X) if ƒ is injective.

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