GATE 2023 MA – Question 32

Functional Analysis · Hilbert Spaces · 1 mark · Numerical answer

On real L²[−1,1], F(f)=∫₋₁¹f(x)x²dx. With the usual L² norm, find 5‖F‖².

Show answer and explanation

Correct answer: 2 to 2

Explanation

The representing vector is x². Cauchy–Schwarz gives ‖F‖=‖x²‖₂, attained by its normalized multiple. Thus ‖F‖²=∫₋₁¹x⁴dx=2/5, and **5‖F‖²=2**.

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