GATE 2021 MA – Question 43

Functional Analysis · Hilbert Spaces · 2 marks · Multiple choice

In real L²[−1,1] with its usual norm, let M={f:∫₋₁¹f=0} and f(x)=x². Find inf_(g∈M)‖f−g‖.

Show answer and explanation

Correct answer: (A) √2/3

Explanation

M is the orthogonal complement of constant 1. The normal projection of x² is [∫x²/∫1]·1=(1/3)·1. Its norm is (1/3)√2, giving distance **√2/3, A**.

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