Mathematics · Real Analysis
Let C 0 (ℝ) be the space of all continuous functions on ℝ such that lim x→±∞ f(x) = 0, equipped with the uniform convergence norm ||.||. Let (f n ) be a sequence in C 0 (ℝ) and f ∈ C 0 (ℝ). Which of the following statements is correct?
Verified Solution4 OptionsUpdated: October 2026👁️1 View
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- 1Neither of the two statements implies the other
- 2f n → f uniformly on compact sets if and only if ||f n - f|| → 0
- 3If ||f n - f|| → 0, then f n → f uniformly on compact sets✓
- 4If f n → f uniformly on compact sets, then ||f n - f|| → 0
Correct answer
If ||f n - f|| → 0, then f n → f uniformly on compact sets
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