Let C[0,1] be the real vector space of real-valued continuous functions equipped with the supremum 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. Which of the following statements is true?

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Mathematics · Functional Analysis

Verified Solution4 OptionsUpdated: October 2026👁️1 View

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Correct answer

The set {f ∞ C[0,1] : T(f) = f} is a singleton set.

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Answer 1

||Tn|| → &inf; as n → &inf;.

Answer 2

The set {f ∞ C[0,1] : T(f) = f} is a singleton set.

Answer 3

There exists α ∞ (0,1) such that for all f, g ∞ C[0,1], ||T2(f) - T2(g)|| ≤ α||f - g||.

Answer 4

There exists α ∞ (0,1) such that for all f, g ∞ C[0,1], ||T(f) - T(g)|| ≤ α||f - g||.

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