# Functional Analysis Problems

## Transcribed Text

1. Let f : [−a, a] → R be a continuous function where a > 0. If f satises that Z a −a f(x)g(x)dx = 0 for every integrable even function g : [−a, a] → R, show that f is an odd function. 2. Let f : [0, 1] → R be a dierentiable function with f(0) = 0 and f 0 (x) ∈ (0, 1), x ∈ (0, 1). Show that Z 1 0 f(x)dx!2 > Z 1 0 (f(x))3 dx (Hint: Show that the function F(x) = (R x 0 f(t)dt) 2 − R x 0 (f(t))3dt is increasing.) 3. For what values of k the improper integral Z ∞ 2 dx x k ln x converges? 4. Apply Bolzano-Cauchy Criterion to show that the improper integral Z ∞ 0 sin x x dx converges. 5. Let f : [a, b] → R be an integral function. Dene f+(x) = ( f(x) if f(x) ≥ 0, 0 if f(x) < 0, and f− = ( 0 if f(x) ≥ 0, −f(x) if f(x) < 0. Prove that f+, f− : [a, b] → R are integrable and Z b a f(x)dx = Z b a f+(x)dx − Z b a f−(x)dx. 6. Let f : [a, b] → R be a Riemann integrable function such that ∃α > 0 3 ∀x ∈ [a, b], f(x) ≥ α. Show that the function g(x) = 1 f(x) is also Riemann integrable on [a, b]. 7. Compute the following indenite integral Z dx 1 + x 4 18. Let f : (0, 1) → R be a continuous function such that the improper integral Z π 0 f(sin x)dx converges. Show that Z π 0 xf(sin x)dx = π Z π 2 0 f(sin x)dx 1. Solve the following problems a) Let f : [a,∞) → R be a function such that the improper integral Z ∞ a f(x)dx converges. Assume the existence of the (nite) limit limx→∞ f(x) = α Show that α = 0. b) Assume that f : [a,∞) → R be a continuously dierentiable function such that the improper integrals Z ∞ a f(x)dx and Z ∞ a f 0 (x)dx converge. Show that limx→∞ f(x) = 0. 2. Compute the following improper integrals Z ∞ 0 xe−αx cos(βx)dx, Z ∞ 0 xe−αx sin(βx)dx. 3. Solve the following problems a) Compute improper integral: Z π 2 0 x cot xdx (Hint: Use the integration by parts). b) Compute improper integral: Z 1 0 arcsin x x dx. (Hint: Use the integration by parts).

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