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GATE 2024 MA – Question 57

Functional Analysis · Hilbert Spaces · 2 marks · Numerical answer

In real L²[−1,1], ‖f‖²=18/5, ∫f=2 and ∫xf=0. P₁ is the subspace of polynomials of degree at most one. Find inf_(g∈P₁)‖f−g‖² to two decimals.

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Correct answer: 1.59 to 1.61

Explanation

The basis 1,x is orthogonal. Projection coefficients are ⟨f,1⟩/‖1‖²=2/2=1 and ⟨f,x⟩/‖x‖²=0. Thus the projection is constant 1 with squared norm 2. Pythagoras gives minimum squared error 18/5−2=**1.60**.

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