Mathematics · Real Analysis
Let C[0,1] be the ℝ-vector space of real-valued continuous functions equipped with the norm ||f|| = supx ∈ [0,1]|f(x)|. Let T: C[0,1] → C[0,1] be defined as T(f)(x) = ∫0x f(t) dt, for x ∈ [0,1]. Let Tn = T º T º ... º T (n times). Which of the following statements is true?
Verified Solution4 OptionsUpdated: October 2026👁️1 View
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- 1||Tn|| → ∞ as n → ∞.
- 2The set {f ∈ C[0,1] : T(f) = f} contains infinitely many functions.
- 3There exists α ∈ (0,1) such that for all f, g ∈ C[0,1], ||T2(f) - T2(g)|| ≤ α||f - g||✓
- 4There exists α ∈ (0,1) such that for all f, g ∈ C[0,1], ||T(f) - T(g)|| ≤ α||f - g||
Correct answer
There exists α ∈ (0,1) such that for all f, g ∈ C[0,1], ||T2(f) - T2(g)|| ≤ α||f - g||
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