Mathematics · Functional Analysis
Consider the following two spaces: X = (C[-1, 1], ||·||∞), the space of all real-valued continuous functions defined on [-1, 1] equipped with the norm ||f||∞ = supt ∈ [-1, 1] |f(t)|. Y = (C[-1, 1], ||·||2), the space of all real-valued continuous functions defined on [-1, 1] equipped with the norm ||f||2 = (∫-11 |f(t)|2 dt)1/2. Let W be the linear span over ℝ of all the Legendre polynomials. Then, which one of the following statements is correct?
Verified Solution4 OptionsUpdated: October 2026👁️1 View
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- 1W is dense neither in X nor in Y
- 2W is dense in both X and Y✓
- 3W is dense in Y but not in X
- 4W is dense in X but not in Y
Correct answer
W is dense in both X and Y
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